# Crossing indices, traits and the principle of indistinguishability

Igor Nikonov111The author was supported by the Russian Foundation for Basic
Research (grant No. 19-51-51004-NIF-a).

###### Abstract

A (weak chord) index is a function on the crossings of knot diagrams such
that: 1) the index of a crossing does not change under Reidemeister moves; 2)
crossings which can be paired by a second Reidemeister move have the same
index. We show that one can omit the second condition in the case of the
universal index. As a consequence, we get the following principle of
indistinguishability for classical knots: crossings of the same sign in a
classical knot diagram can not be distinguished by any inherent property.

## Introduction

The conventional combinatorial approach to knot theory looks at knots as
diagrams considered up to special transformations — Reidemeister moves. Many
constructions of knot invariants rely on combinatorial elements of diagrams —
arcs and crossings. But what if we could distinguish crossings of knot
diagrams and divide them into classes and tribes? For example, discern a loop
crossing which appears in a first Reidemeister move and a crossing which
survived the reduction to the minimal diagram (Fig. 1).

Figure 1: Crossings in a knot diagram

For virtual knots, this idea was excellently realized by V.O. Manturov who
created the parity theory [31] and then applied it to strengthen knot
invariants, to prove minimality theorem and to construct (counter)examples
[32, 33, 34, 35], see also [17, 18, 19, 36, 39].

Despite all achievements of the parity theory in the theory of virtual knots,
it has one flow: the parity theory gives nothing to classical knots [18,
Corollary 4.3]. The reason is any parity on a knot is derived from the
homology of the space the knot lies in, and the plane where the classical
knots live, has trivial homology.

After this failure with parity we can try to weaken the parity axioms and
consider another type of labelling of crossings — index. The first appearance
of an index can be found in V. Turaev’s work [41] on strings (flat knots).
After the paper of A. Henrich [15], the index appeared in works of several
authors [3, 20, 28] and others. Z. Cheng [6] gave an axiomatic description of
the index. M. Xu [43] found the most general formulation of index which he
called _weak chord index_.

The index polynomial has become a very efficient tool in the virtual knot
theory. May be, it will be useful for study of classical knots. But, alas!
There is no nontrivial indices on classical knots [40, Corollary 5].

This paper is a sequel of [40]. Here we make the final step and show that
there can not be nontrivial invariant labels (except the sign, surely) on
crossings of classical knot diagrams. This fact can be formulated as the
following _principle of indistinguishability_ :

> Crossings of the same sign in a classical knot diagram can not be
> distinguished by any inherent property.

The mechanism behind the indistinguishability principle can be expressed as
the _substitution principle_ :

> Given two crossing of the same sign in a classical knot diagram, one of them
> can substitute for the other in the diagram.

The exact formulation of the principles are given in Theorems 4 and 5.

A demonstration how the substitution principle works is shown in Fig. 2.

Figure 2: A crossing substitutes for another one

The paper is organized as follows. In Section 1 we define categories of
tangle, link and knot diagrams. In Section 2 we introduce the notions of
(signed) index, index polynomial and the universal index. Section 3 contains
the central result of the paper: we show that the universal trait (the
invariant labelling of crossing which contains the most information on them)
is the universal signed index (Theorem 1). In Section 4 we combine the central
result with the description of the universal index from [40] and formulate the
principles of indistinguishability and substitution. We conclude the paper
with the appendix where we list some known examples of index.

##  1 Knot theories

We start with definitions of virtual knots and related objects.

######  Definition 1.

Let $F$ be an oriented compact connected surface with the boundary $\partial
F$ (perhaps empty). An _oriented tangle_ in the thickened surface
$F\times(0,1)$ is an embedding $T\colon M\to F\times(0,1)$ of an oriented
compact $1$-dimensional manifold $M$ such that $T(\partial M)\subset\partial
F\times(0,1)$ and $T$ is transversal to $\partial F\times(0,1)$. The image of
the map $T$ will be also called a tangle.

A connected component of a tangle whose boundary is empty is called a _closed
component_ , otherwise, the components is called _long_.

If a tangle consists of one closed component then it is a _knot_ ; a tangle
consisting of one long component is a _long knot_ ; a tangle with several
component which are all closed, is a _link_.

We consider tangles up to isotopies of $F\times(0,1)$ with the boundary
$\partial F\times(0,1)$ fixed.

A _tangle diagram_ $D$ is the image of a tangle $T$ in general position under
the natural projection $F\times(0,1)\to F$. Combinatorially, a tangle diagram
$D$ is an embedded graph in $F$ with vertices of valency $4$ (called
_crossings_) and vertices of valency $1$ (forming the _boundary_ $\partial D$
of the diagram $D$) such that $D\cap\partial F=\partial D$ and each vertex of
valency $4$ carries the structure of under-overcrossing (Fig. 3).

Figure 3: A tangle diagram

Any tangle diagrams which correspond to isotopical tangles can be connected by
a sequence of isotopies of the surface $F$ identical on $\partial F$, and
_Reidemeister moves_ (Fig. 4).

1em

Figure 4: Reidemeister moves

Let $D$ be a tangle diagram. Denote the set of its crossings by
${\mathcal{V}}(D)$. We assume that all tangle diagrams are oriented, hence,
any crossing $v\in{\mathcal{V}}(D)$ has a _sign_ (Fig. 5).

Figure 5: The sign of a crossing

_Flat tangles in the surface $F$_ can be defined as immersions of an oriented
compact 1-manifold $T$ into the thickened surface $F\times(0,1)$ such that
$\partial T=T\cap\partial F\times(0,1)$ up to homotopies fixed on the boundary
$\partial T$. Combinatorially, one defines flat tangles as equivalence classes
of tangle diagrams modulo isotopies, Reidemeister moves and crossing changes
(Fig. 6).

Figure 6: Change of a crossing

Flat tangle diagrams are obtained from tangle diagrams by forgetting under-
overcrossing structure (Fig. 7).

Figure 7: A flat tangle diagram

_Oriented virtual tangles_ $\mathcal{T}$ are equivalence classes of pairs
$(F,T)$ where $F$ is an oriented closed surface and $T\subset F\times(0,1)$ is
an oriented tangle in the thickening of $F$, considered modulo isotopies,
diffeomorphisms and stabilization/destabilization operations [2]. The
stabilization operations are: attaching a (thickened) 2-handle distinct from
$T$ to $F\times(0,1)$, sealing a boundary component distinct from $T$ with a
(thickened) disk, and gluing a (thickened) handle distinct from the tangle
boundary $\partial T$ to a boundary component of $F\times(0,1)$, see Fig. 8.

Figure 8: The three stabilization operations

_Virtual tangle diagrams_ are pairs $(F,D)$ where $F$ is a surface and
$D\subset F$ is a tangle diagram (a general projection of $T$ for some
representative $(F,T)$ of $\mathcal{T}$). One can consider virtual tangles as
the equivalence classes of diagrams up to surface isotopies (fixed on the
boundary), Reidemeister moves, diffeomorphisms of pairs $(F,D)$ and
(de)stabilizations.

A _flat tangle diagram_ is a pair $(F,D)$ where $F$ is an oriented compact
connected surface and $D\subset F$ is an embedded graph with vertices of
valency $4$ and vertices of valency $1$ (forming the boundary $\partial D$ of
the diagram $D$) such that $D\cap\partial F=\partial D$. A _flat tangle_ is an
equivalence class of flat link diagrams modulo isotopies of the surface,
Reidemeister moves, orientation preserving diffeomorphisms of pairs $(F,D)$
and (de)stabilizations.

Another approach to virtual (flat) tangles uses plane virtual diagrams [27].

Fix a disk $\mathbb{D}$ in the plane ${\mathbb{R}}^{2}$.

A virtual diagram is an embedded graph $D\subset\mathbb{D}$ which has vertices
of valency four (crossings) and one (boundary vertices). The set $\partial D$
of the vertices of valency one coincides with $D\cap\partial\mathbb{D}$. Each
vertex of valency $4$ of the graph $D$ is marked as either classical of
virtual vertex. Virtual crossings of a virtual diagram are drawn circled (see
Fig. 9). A diagram without virtual crossings is classical.

Figure 9: Virtual tangle diagram with three classical and two virtual
crossings

Moves of virtual diagrams include classical Reidemeister moves ($R1$, $R2$,
$R3$), detour moves ($DM$) and _boundary moves_ (Fig. 10). A detour move
replaces any diagram arc, which has only virtual crossings, with a new arc,
which has the same ends and contains only virtual crossings. A boundary move
interchanges two neighboring boundary vertices of a virtual tangle diagram and
adds a virtual crossing. An equivalence class of virtual diagram modulo moves
is called a virtual tangle.

Figure 10: Detour and boundary moves

A classical tangle is a tangle which has at least one classical diagram.

If one admits crossing switch transformations (Fig. 6) of virtual diagrams,
i.e. neglects the over-undercrossing structure, one gets the theory of flat
tangles.

Thirdly, oriented virtual tangles can be defined by means of Gauss diagrams
[13]. The Gauss diagram $G=G(D)$ of a virtual tangle diagram $D$ is a chord
diagram on an oriented 1-dimensional manifold $G$ which is immersed onto $D$.
The chords of the chord diagram correspond to the classical crossings of $D$
(i.e. the double points of the immersion), see Fig. 11. The chords carry an
orientation (from over-crossing to under-crossing) and the sign of the
crossings, see Fig. 5.

Figure 11: The Gauss diagram of the virtual tangle in Fig. 9

Classical Reidemeister moves induce transformations of Gauss diagrams (see
Fig. 12). Virtual tangles are the equivalence classes of Gauss diagrams modulo
the induced Reidemeister moves.

Figure 12: Reidemeister moves on Gauss diagrams

######  Theorem.

The three definitions of virtual tangles are equivalent.

The proof of the theorem reproduces the corresponding proof for virtual links
[2, 13].

The factorization by crossing change and virtualization moves (Fig. 13) leads
to the theory of free tangles.

Figure 13: Virtualization move

Crossing change and virtualization moves induces transformations on Gauss
diagrams (Fig. 14). Therefore, the theory of free tangles which is obtained by
factorization by these transformations, is the theory of chord diagrams
without orientation and labels on the chords. The moves of free tangles are
the moves in Fig. 12 after one has wiped the arrowheads and the signs.

Figure 14: Crossing change (left) and virtualization (right) of a Gauss
diagram

###  1.1 Diagram categories

Let $\mathcal{K}$ be a knot. We shall use the notion of ‘knot’ in one of the
following situations:

  * •

virtual knot (long knot, link or tangle);

  * •

flat knot (long knot, link or tangle);

  * •

free knot (long knot, link or tangle);

  * •

knot (long knot, link or tangle) in a fixed surface;

  * •

flat knot (long knot, link or tangle) in a fixed surface;

  * •

classical knot (long knot, link or tangle).

We always assume that the components of tangle are numbered and oriented.

######  Definition 2.

The category of diagrams $\mathfrak{K}=\mathfrak{K}(\mathcal{K})$ of the knot
$\mathcal{K}$ is the small category whose objects are the diagrams of
$\mathcal{K}$ and morphisms are (formal) compositions of elementary morphisms.
By an elementary morphism we mean

  * •

an isotopy of a diagram;

  * •

an isomorphism between diagrams (in case of virtual, flat or free tangles);

  * •

a Reidemeister move.

Given a morphism $f\colon D\to D^{\prime}$ in $\mathfrak{K}$, generally there
is no bijection between the sets of crossings ${\mathcal{V}}(D)$ and
${\mathcal{V}}(D^{\prime})$ of the diagrams because the number of crossings of
a diagram may change under Reidemeister moves. But there is a bijection for
the crossings which survives the morphism: there are subsets
$\bar{\mathcal{V}}(D)\subset{\mathcal{V}}(D)$ and
$\bar{\mathcal{V}}(D^{\prime})\subset{\mathcal{V}}(D^{\prime})$ which are in
an one-to-one correspondence
$\bar{\mathcal{V}}(D)\to\bar{\mathcal{V}}(D^{\prime})$. We denote this
correspondence by $f_{*}$ and the subsets by $dom(f_{*})=\bar{\mathcal{V}}(D)$
and $im(f_{*})=\bar{\mathcal{V}}(D^{\prime})$.

Usually, one works not with one knot or tangle but with knots from a certain
class (e.g. virtual knots, virtual links, free knots etc.). In this case, we
have to deal with diagrams of all such knots, that is work with diagram
category which is the union of the categories of diagrams on the knots. We
call this “large” categories _knot theories_.

######  Definition 3.

Given a tangle $T$, its _type_ $t(T)$ is the pair $(m,n)$ where $m$ is the
number of closed components in $T$ and $n$ is the number of the long
components.

For a compact oriented surface $F$, the _theory of tangles of type $(m,n)$
with the boundary $B$ in the surface $F$_ is the diagram category

$\mathfrak{T}_{m,n}(F,B)=\bigsqcup_{T\colon t(T)=(m,n),\ \partial
T=B}\mathfrak{K}(T),$

where $B\subset\partial F$, $|B|=2n$.

The _theory of flat tangles of type $(m,n)$ with the boundary $B$ in the
surface $F$_ is the union

$\mathfrak{FT}_{m,n}(F,B)=\bigsqcup_{T\mbox{\scriptsize\ is flat}\colon
t(T)=(m,n),\ \partial T=B}\mathfrak{K}(T).$

The _theory of virtual tangles of type $(m,n)$_ is the diagram category

$\mathfrak{VT}_{m,n}=\bigsqcup_{T\mbox{\scriptsize\ is a virtual tangle}\colon
t(T)=(m,n)}\mathfrak{K}(T),$

the _theory of flat tangles of type $(m,n)$_ is the diagram category

$\mathfrak{FT}_{m,n}=\bigsqcup_{T\mbox{\scriptsize\ is a flat tangle}\colon
t(T)=(m,n)}\mathfrak{K}(T),$

and the _theory of free tangles of type $(m,n)$_ is the diagram category

$\mathfrak{FrT}_{m,n}=\bigsqcup_{T\mbox{\scriptsize\ is a free tangle}\colon
t(T)=(m,n)}\mathfrak{K}(T).$

As special cases, we get the _theories of classical knots_
$\mathfrak{CK}=\mathfrak{T}_{1,0}(S^{2},\emptyset)$, _virtual knots_
$\mathfrak{VK}=\mathfrak{VT}_{1,0}$, _long virtual knots_
$\mathfrak{LVK}=\mathfrak{VT}_{0,1}$, _virtual links with $m$ components_
$\mathfrak{VL}_{m}=\mathfrak{VT}_{m,0}$ etc.

######  Remark 1.

Knot theories are connected by natural transformations. For example, we have a
sequence of projections
$\mathfrak{T}_{m,n}(F,B)\to\mathfrak{VT}_{m,n}\to\mathfrak{FT}_{m,n}\to\mathfrak{FrT}_{m,n}$
and $\mathfrak{T}_{m,n}(F,B)\to\mathfrak{FT}_{m,n}(F,B)$ from (virtual)
tangles to flat tangles and then to free tangles. In particular, that means
invariants of free tangles can be lifted to invariants of flat tangles, and
invariants of flat tangles are lifted to invariants of virtual tangles.

We will use the name _diagram category_ both for category of diagrams of a
given tangle and for knot theories defined above.

##  2 Indices

The notion of index was axiomatized by Z. Cheng [6] in the spirit of
Manturov’s parity axioms [31]. M. Xu [43] found the general conditions for an
index to define an invariant polynomial and introduced the notion a weak chord
index. See also [6, 7, 9] for other index axiomatics and examples of chord
indices.

######  Definition 4.

Let $\mathfrak{K}$ be a diagram category. An _index with coefficients in a set
$I$ on the diagram category $\mathfrak{K}$_ is a map $\iota$ which assigns
some value $\iota(v)\in I$ to each crossing $v\in{\mathcal{V}}(D)$ in each
diagram $D\in\mathfrak{K}$ of the diagram category and possesses the following
properties:

  * (I0)

for any elementary morphism $f\colon D\to D^{\prime}$ in $\mathfrak{K}$ and
any crossings $v\in dom(f_{*})$ one has $\iota(v)=\iota(f_{*}(v))$;

  * (I2)

$\iota(v_{1})=\iota(v_{2})$ for any crossings
$v_{1},\,v_{2}\in{\mathcal{V}}(D)$ to which a decreasing second Reidemeister
move can be applied.

Let $S$ be a set with an involution $\ast\colon S\to S$. A _signed index_ with
coefficients in the set $S$ on the diagram category $\mathfrak{K}$ is a map
$\sigma\colon\bigsqcup_{D\in\mathfrak{K}}{\mathcal{V}}(D)\to S$ which
satisfies the property (I0) and the property

  * (I2+)

for any crossings $v_{1},\,v_{2}\in{\mathcal{V}}(D)$ to which a decreasing
second Reidemeister move can be applied, one has
$\sigma(v_{1})=\sigma(v_{2})^{\ast}$.

######  Remark 2.

1\. For a (signed) index, the property (I0) holds for all morphisms of the
diagram category $\mathfrak{K}$.

2\. An index $\iota$ with coefficients in a set $I$ can be considered as a
signed index if one takes the trivial involution on $I$: $x^{*}=x$ for all
$x\in I$.

3\. If $\sigma$ is a signed index with coefficients is a set $S$ with an
involution $\ast$ then the composition $\bar{\sigma}=p\circ\sigma$ of $\sigma$
with the projection $p\colon S\to S/\ast$, where $S/\ast$ is the quotient set
by the involution action, is an index with coefficients in the set $S/\ast$.

######  Example 1.

The sign function is a signed index with coefficients in
${\mathbb{Z}}_{2}=\\{-1,+1\\}$ for the diagram categories
$\mathfrak{T}_{m,n}(F,B)$ and $\mathfrak{VT}_{m,n}$ of tangles in a surface
and virtual tangles and all their diagram subcategories.

Usually when an index is mentioned, one means the ${\mathbb{Z}}$-valued index
appeared in [3, 28] or a ${\mathbb{Z}}$-valued signed index introduced in [41,
15, 20, 12]. In order to differ it from the general notion of index, we will
call it the _Gaussian index_ by analogy with the Gaussian parity [19].

######  Example 2 (Gaussian index).

Let $K$ be a knot in a surface $F$ or a virtual knot. For any diagram $D$ of
$K$ and any crossing $v\in{\mathcal{V}}(D)$, the oriented smoothing at $v$
splits the component $D_{i}$ into two the _left half_ $D^{l}_{v}$ and the
_right half_ $D^{r}_{v}$ (Fig. 15). We define also the _signed halves_
$D^{\pm}_{v}$ by the formula $D^{sgn(v)}_{v}=D^{l}_{v}$ and
$D^{-sgn(v)}_{v}=D^{r}_{v}$.

Figure 15: The left and the right halves of the diagram

Then define the _Gaussian index_ of the crossing as the intersection number of
two cycles: $Ind(v)=D\cdot D^{+}_{v}\in{\mathbb{Z}}$. Then the map $Ind$ obeys
the conditions (I0) and (I2). It is an index with coefficients in
${\mathbb{Z}}$ on diagram categories of virtual knots and knots in a fixed
surface.

The index $n$ defined by V. Turaev [41] can be obtained by the formula
$n(v)=D\cdot D^{l}_{v}$,. In fact, it is a signed index with coefficients in
${\mathbb{Z}}$ on diagram categories of virtual and flat knots and (flat)
knots in a fixed surface.

For virtual knots, the indices $Ind$ and $n$ are connected by the formula
$Ind(v)=sgn(v)\cdot n(v)$.

In the presence of the sign function there is a correspondence between indices
and signed indices as the previous example shows. Let us describe it in a more
general setting.

######  Proposition 1.

Let $\mathfrak{K}$ be a diagram category $\mathfrak{T}_{m,n}(F,B)$,
$\mathfrak{VT}_{m,n}$ or any subcategory of those.

  1. 1.

If $\sigma$ is a signed index with coefficients in a set $S$ with an
involution $\ast$. Then the map $\hat{\sigma}$ such that
$\hat{\sigma}(v)=\sigma(v)^{sgn(v)}$, where $v\in{\mathcal{V}}(D)$,
$D\in\mathfrak{K}$, and $x^{+1}=x$ and $x^{-1}=x^{\ast}$ for $x\in S$, is an
index with coefficients in the set $S$ on the diagram category $\mathfrak{K}$.

  2. 2.

If $\iota$ is an index with coefficients in a set $S$ with an involution
$\ast$. Then the map $\hat{\iota}$ such that
$\hat{\iota}(v)=\iota(v)^{sgn(v)}$, $v\in{\mathcal{V}}(D)$,
$D\in\mathfrak{K}$, is a signed index with coefficients in the set $S$ on the
diagram category $\mathfrak{K}$.

  3. 3.

If $\iota$ is an index with coefficients in a set $I$ on $\mathfrak{K}$. Then
the map $\tilde{\iota}$ such that $\tilde{\iota}(v)=(\iota(v),sgn(v))\in
I\times\\{-1,+1\\}$ is a signed index on $\mathfrak{K}$ with coefficients in
the set $I\times\\{-1,+1\\}$ with the involution
$(x,\epsilon)^{\ast}=(x,-\epsilon)$, $x\in I$, $\epsilon\in\\{-1,+1\\}$.

Let us describe how a (signed) index relates first Reidemeister move. A first
Reidemeister move can create a loop crossing of one of the four types in Fig.
16.

Figure 16: Types of loops

######  Proposition 2.

Let $\mathfrak{K}=\mathfrak{K}(\mathcal{T})$ be the category of diagrams of a
tangle $\mathcal{T}=T_{1}\cup\cdots\cup T_{l}$ with $l$ component, $\iota$ be
an index on $\mathfrak{K}$ with coefficients in a set $I$, and $\sigma$ be a
signed index on $\mathfrak{K}$ with coefficients in a set with involution $S$.
Then there exist values $\iota_{i}^{l\pm},\iota_{i}^{r\pm}\in I$ and
$\sigma_{i}^{l\pm},\sigma_{i}^{r\pm}\in S$, $i=1,\dots,l$, such that

  1. 1.

any loop crossing $v$ of type $l+$ (corr., $l-$, $r+$, $r-$) in a component
$T_{i}$ has the index value $\iota_{i}^{l+}$ (corr.,
$\iota_{i}^{l-},\iota_{i}^{r+},\iota_{i}^{r-}$) and the signed index value
$\sigma_{i}^{l+}$ (corr., $\sigma_{i}^{l-},\sigma_{i}^{r+},\sigma_{i}^{r-}$);

  2. 2.

$\iota_{i}^{l+}=\iota_{i}^{r-}$, $\iota_{i}^{l-}=\iota_{i}^{r+}$, and
$\sigma_{i}^{r-}=(\sigma_{i}^{l+})^{\ast}$,
$\sigma_{i}^{r+}=(\sigma_{i}^{l-})^{\ast}$, $i=1,\dots,l$.

###### Proof.

Due to Remark 2 it is enough to give a proof for the signed index.

Let $v\in{\mathcal{V}}(D_{1})$ be any loop index of type $l+$ in a component
$T_{i}$, and $w\in{\mathcal{V}}(D_{2})$ be a loop crossing of type $r-$ in
$T_{i}$, $D_{1},D_{2}\in\mathfrak{K}$. We show that
$\sigma(w)=\sigma(v)^{\ast}$.

Since $D_{1}$ and $D_{2}$ are diagrams of one tangle $\mathcal{T}$, there is a
morphism $f\colon D_{2}\to D_{1}$. We modify the morphism $f$ as follows: we
add a double $w^{\prime}$ of the crossing $w$ in the same component of the
diagram $D_{2}$ and use the double instead of $w$ in the Reidemeister moves of
the morphism $f$, and keep the loop with the crossing $w$ somewhere on the
diagrams. Thus, we get a sequence of diagrams $D_{2}\stackrel{{\scriptstyle
R_{1}}}{{\rightarrow}}D_{2}^{\prime}\stackrel{{\scriptstyle
f^{\prime}}}{{\rightarrow}}D^{\prime}_{1}$ where $D_{2}^{\prime}$ is the
diagram with the double added, $f^{\prime}$ is the morphism $f$ with added
loop $w$, and $D^{\prime}_{1}$ differs from $D_{1}$ by a first Reidemeister
move which adds the loop crossing $w^{\prime\prime}=f^{\prime}_{*}(w)$.

We can move the crossing $f^{\prime}_{*}(w)$ to the crossing $v$ and assume
that these crossings are neighboring in the diagram $D_{1}^{\prime}$. Then
move the crossing $w^{\prime\prime}$ along the component as shown in Fig. 17.
Then by the property $(I2+)$ $\sigma(w^{\prime\prime})=\sigma(v)^{\ast}$. By
the property $(I0)$
$\sigma(w)=\sigma(f^{\prime}_{*}(w))=\sigma(w^{\prime\prime})=\sigma(v)^{\ast}$.

Figure 17: Pairing loops

Let $v\in{\mathcal{V}}(D)$ and $v^{\prime}\in{\mathcal{V}}(D^{\prime})$ be two
loop crossings of the type $l+$. Create in some diagram
$D^{\prime\prime}\in\mathfrak{K}$ a loop crossing $w$ of type $r-$ in the
$i$-th component. Then
$\sigma(v^{\prime})=\sigma(w)^{\ast}=(\sigma(v)^{\ast})^{\ast}=\sigma(v)$.
Thus, all the loop crossings of type $l+$ in the component $T_{i}$ have the
same signed index value $\sigma_{i}^{l+}\in S$. The proof for other types is
analogous.

The equality $\sigma_{i}^{r-}=(\sigma_{i}^{l+})^{\ast}$ was proved above.
Analogously, $\sigma_{i}^{r+}=(\sigma_{i}^{l-})^{\ast}$. ∎

######  Remark 3.

  1. 1.

The values $\iota_{i}^{l\pm},\iota_{i}^{r\pm}\in I$ and
$\sigma_{i}^{l\pm},\sigma_{i}^{r\pm}\in S$ can coincide;

  2. 2.

if $\mathcal{T}$ is a flat tangle and $\iota$ is an index, and $\sigma$ is a
signed index on the category $\mathfrak{K}(\mathcal{T})$ of diagrams of
$\mathcal{K}$, then the crossings don’t have signs, and the loop crossings in
one component of the tangle have one index value
$\iota_{i}=\iota_{i}^{l\pm}=\iota_{i}^{r\pm}$, and two signed index values
$\sigma_{i}^{l}$ and $\sigma_{i}^{r}=(\sigma_{i}^{l})^{\ast}$;

  3. 3.

if $\mathcal{K}$ is a free tangle then all the loop crossings on the same
component have the same index or signed index value.

  4. 4.

if one considers a knot theory $\mathfrak{K}$ which consists of diagram
categories $\mathfrak{K}(T)$ of different tangles, the loop indices can be
individual for each tangle $T$: $\iota_{i}^{l\pm}(T),\iota_{i}^{r\pm}(T)$.

Some indices may have one loop index value for different tangles. For example,
for the ${\mathbb{Z}}$-valued indices $Ind$ and $n$, the loop index values are
all equal to zero.

Thus, the first Reidemeister move can be localized and isolated by neglecting
the index values $\iota^{l\pm},\iota^{r\pm}$ ($\sigma^{l\pm},\sigma^{r\pm}$).
We can use this fact and define a tangle invariant (the index polynomial).

###  2.1 Index polynomial

######  Definition 5.

Let $\mathfrak{K}$ be a diagram category and $\sigma$ be a signed index on
$\mathfrak{K}$ with coefficients in a set $S$. Let $L\subset S$ be the set of
loop index values, i.e.
$L=\bigcup_{i}\\{\sigma_{i}^{l\pm},\sigma_{i}^{r\pm}\\}$ if $\mathfrak{K}$ is
the category of diagrams of a tangle $T=\bigcup_{i}T_{i}$, or
$L=\bigcup_{T\in{\mathfrak{K}},\,i}\\{\sigma_{i}^{l\pm}(T),\sigma_{i}^{r\pm}(T)\\}$
if $\mathfrak{K}$ is a knot theory. Note that $L^{\ast}=L$.

Consider the abelian group $A_{\sigma}={\mathbb{Z}}[S\setminus L]/\langle
1\cdot x+1\cdot x^{\ast}\mid x\in S\setminus L\rangle$.

Let $T$ be a tangle presented by a diagram $D\in\mathfrak{K}$. The _linking
invariant_ (or _(odd) index polynomial_) $lk_{\sigma}(T)$ of the tangle $T$ is
the element

$lk_{\sigma}(T)=\sum_{v\in{\mathcal{V}}(D)\,\colon\,\sigma(v)\not\in
L}1\cdot\sigma(v)\in A_{\sigma}.$ (1)

For an index $\iota$ on $\mathfrak{K}$ with coefficients in a set $I$ we
define the _linking invariant_ $lk_{\iota}(T)$ as $lk_{\tilde{\iota}}(T)$ if
the category $\mathfrak{K}$ has the sign function, and as $lk_{\iota}(T)$ if
$\mathfrak{K}$ is a diagram category of flat or free tangles. In the latter
case we consider the trivial involution on the set $I$.

######  Proposition 3.

The index polynomial $lk_{\sigma}$ is an invariant of tangles in the diagram
category $\mathfrak{K}$, i.e. the value $lk_{\sigma}(T)$ does not depend on
the choice of a diagram $D$ representing the tangle $T$.

###### Proof.

It is enough to prove that for any elementary morphism $f\colon D_{1}\to
D_{2}$ in $\mathfrak{K}$ the values of $lk_{\sigma}$ for $D_{1}$ and $D_{2}$
coincide. When $f$ does not change the number of crossing, this follows from
the property (I0). If $f$ is a second Reidemeister move then the equality
follows from the property (I2). We secure the invariance under first
Reidemeister moves by excluding the loop index values. ∎

######  Remark 4.

1\. The group $A_{\sigma}$ is a direct sum of the cyclic groups ${\mathbb{Z}}$
(one summand for each pair $\\{x,x^{\ast}\\}$ such that $x\in S\setminus L$
and $x\neq x^{\ast}$) and ${\mathbb{Z}}_{2}$ (one summand for each self-dual
element $x=x^{\ast}\in S\setminus L$).

For an index $\iota$ with coefficients in a set $I$, the group
$A_{\iota}=A_{\tilde{\iota}}$ is isomorphic to ${\mathbb{Z}}[I\setminus L]$ in
the presence of the sign function on the crossings (one maps $(x,\epsilon)$ to
$\epsilon\cdot x$), and $A_{\iota}={\mathbb{Z}}_{2}[I\setminus L]$ for diagram
categories of flat or free tangles.

2\. Let $\iota$ be an index on a diagram category $\mathfrak{K}$ with
coefficients in a set $I$, and the crossings of the diagrams in $\mathfrak{K}$
have signs. Using the isomorphism $A_{\iota}\simeq{\mathbb{Z}}[I\setminus L]$
we can write the formula the odd index polynomial of a tangle $T$ as follows

$lk_{\iota}(T)=\sum_{v\in{\mathcal{V}}(D)\,\colon\,\iota(v)\not\in
L}sgn(x)\cdot\iota(v)\in{\mathbb{Z}}[I\setminus L]$ (2)

where $D\in\mathfrak{K}$ is a diagram of the tangle $T$.

The formula (2) can be considered as a generalization of the well-known
formula for the linking number. This justifies the notation $lk$ which we use
for the odd index polynomial.

######  Example 3.

1\. Let $\mathfrak{VK}$ be the theory of virtual knots. Let $Ind$ be the index
with coefficients in ${\mathbb{Z}}$ defined in Example 2. The set of loop
index values is $L=\\{0\\}$. Then the linking invariant of a virtual knot $K$
given by a diagram $D\in\mathfrak{VK}$

$lk_{Ind}(K)=\sum_{v\in{\mathcal{V}}(D)\,\colon\,Ind(v)\neq 0}sgn(v)\cdot
Ind(v)\in{\mathbb{Z}}[{\mathbb{Z}}\setminus\\{0\\}]$

is (up to isomorphism and the multiplier $t$) the odd index polynomial $f(t)$
defined in [3].

2\. Let $\mathfrak{FK}$ be the theory of flat knots. Let $n$ be the index with
coefficients in ${\mathbb{Z}}$ defined in Example 2. The set of loop index
values is $L=\\{0\\}$. The group $A_{n}$ is isomorphic to the group
${\mathbb{Z}}[\mathbb{N}]$ by the formula $x\mapsto sgn(x)\cdot|x|$. Using
this isomorphism, we write the linking invariant of a flat knot $K$ with a
diagram $D\in\mathfrak{FK}$ as

$lk_{n}(K)=\sum_{v\in{\mathcal{V}}(D)\,\colon\,n(v)\neq
0}sgn(n(v))\cdot|n(v)|\in{\mathbb{Z}}[\mathbb{N}].$

The linking invariant $lk_{n}(K)$ coincides with the polynomial invariant
$u(K)(t)\in{\mathbb{Z}}[t]$ defined by Turaev [41].

###  2.2 Universal index

Let $\mathfrak{K}$ be a diagram category. There can be many (signed) indices
on $\mathfrak{K}$. We are interested in the (signed) index which carries the
maximal information about the crossings.

######  Definition 6.

Let $\mathfrak{K}$ be a diagram category. An index $\iota^{u}$ with
coefficients in a set $I^{u}$ on the diagram category $\mathfrak{K}$ is called
the _universal index_ on $\mathfrak{K}$ if for any index $\iota$ on
$\mathfrak{K}$ with coefficients in a set $I$ there is a unique map
$\psi\colon I^{u}\to I$ such that $\iota=\psi\circ\iota^{u}$.

A signed index $\sigma^{u}$ on the diagram category $\mathfrak{K}$ with
coefficients in a set $S^{u}$ with an involution $\ast$ is _universal_ if for
any signed index $\sigma$ on $\mathfrak{K}$ with coefficients in a set with
involution $S$ there is a unique map $\psi\colon S^{u}\to S$ such that
$\psi(x)^{\ast}=\psi(x^{\ast})$ for all $x\in S^{u}$ and
$\sigma=\psi\circ\sigma^{u}$.

######  Remark 5.

The universal index for the diagram categories $\mathfrak{K}(T)$ of tangles
and flat tangles in a surface $F$ was described in [40].

######  Definition 7.

A (signed) index $\sigma$ on a diagram category $\mathfrak{K}$ with
coefficients in a set $S$ is called _reduced_ if $S=im(\sigma)$ where

$im(\sigma)=\\{\sigma(v)\mid v\in{\mathcal{V}}(D),\ D\in\mathfrak{K}\\}.$

######  Proposition 4.

1\. The universal (signed) index on a diagram category is reduced.

2\. Let $\sigma^{u}$ be the universal signed index on a diagram category
$\mathfrak{K}$. Then $\bar{\sigma}^{u}$ (see Remark 2) is the universal index
on $\mathfrak{K}$.

3\. If a diagram category $\mathfrak{K}$ has the sign function (i.e. it is a
subcategory of $\mathfrak{T}_{m,n}(F,B)$ or $\mathfrak{VT}_{m,n}$) and
$\iota^{u}$ is the universal index on $\mathfrak{K}$ then $\tilde{\iota}^{u}$
is the universal signed index on $\mathfrak{K}$.

###### Proof.

1\. Let $\mathfrak{K}$ be a diagram category and $\sigma^{u}$ with
coefficients in a set $S^{u}$ with an involution $\ast$ be the universal
signed index on $\mathfrak{K}$. Let $S^{\prime}=im(\sigma^{u})\subset S$. Then
the co-restriction $\sigma^{\prime}$ of the index $\sigma^{u}$ to the
coefficient set $S^{\prime}$ is a signed index on $\mathfrak{K}$. By
universality there exists a unique $\ast$-map $\psi\colon S^{u}\to S^{\prime}$
such that $\sigma^{\prime}=\psi\circ\sigma^{u}$. Then the composition
$\psi^{\prime}$ of maps
$S^{u}\stackrel{{\scriptstyle\psi}}{{\rightarrow}}S^{\prime}\hookrightarrow
S^{u}$ satisfies the condition $\psi^{\prime}\circ\sigma^{u}=\sigma^{u}$. Then
by universality, the map $\psi^{\prime}$ must coincide with the identity map
$id_{S^{u}}$. Thus, $S^{\prime}=S^{u}$.

2\. Let $\iota^{u}$ with coefficients in a set $I^{u}$ be the universal index
on $\mathfrak{K}$. By universality of $\iota^{u}$ there exists a map
$\psi\colon I^{u}\to S^{u}/\ast$ such that
$\bar{\sigma}^{u}=\psi\circ\iota^{u}$. On the other hand, by universality of
$\sigma^{u}$ there exists a map $\phi\colon S^{u}\to I^{u}$ such that
$\phi(x^{\ast})=\phi(x)^{\ast}=\phi(x)$ for all $x\in S^{u}$ and
$\iota^{u}=\phi\circ\sigma^{u}$. The map $\phi$ induces a well-defined map
$\bar{\phi}\colon S^{u}/\ast\to I^{u}$. Then the maps $\psi$ and $\bar{\phi}$
define an isomorphism of the indices $\iota^{u}$ and $\bar{\sigma}^{u}$.

3\. By universality of $\sigma^{u}$ there is a map $\psi$ from $S^{u}$ to
$I^{u}\times\\{-1,+1\\}$. This map is injective. Indeed, if $\psi(x)=\psi(y)$
then take crossings $v$ and $w$ such that $\sigma^{u}(v)=x$ and
$\sigma^{u}(w)=y$. Since $\iota^{u}(v)=\iota^{u}(w)$, either
$\sigma^{u}(v)=\sigma^{u}(w)$ or $\sigma^{u}(v)=\sigma^{u}(w)^{\ast}$. In the
latter case the crossings $v$ and $w$ would have different signs. Thus,
$\sigma^{u}(v)=\sigma^{u}(w)$. Then $\psi$ is a bijection. ∎

##  3 Traits and indices

The aim of this section is to prove the central result of the paper that one
can weaken the definition of the signed universal index by removing the
condition (I2+). Let us give the necessary definition.

######  Definition 8.

Let $\mathfrak{K}$ be a diagram category. A _trait with coefficients in a set
$\Theta$ on the diagram category $\mathfrak{K}$_ is a map
$\theta\colon\bigsqcup_{D\in\mathfrak{K}}{\mathcal{V}}(D)\to\Theta$ which
satisfies the property (I0): for any elementary morphism $f\colon D\to
D^{\prime}$ in $\mathfrak{K}$ and any crossings $v\in dom(f_{*})$ one has
$\theta(v)=\theta(f_{*}(v))$.

By definition any index or signed index is a trait.

###  3.1 Universal trait

By analogy with the (signed) index case, one can define the universal trait.

######  Definition 9.

Let $\mathfrak{K}$ be a diagram category. An trait $\theta^{u}$ with
coefficients in a set $\Theta^{u}$ on the diagram category $\mathfrak{K}$ is
called the _universal trait_ on $\mathfrak{K}$ if for any trait $\theta$ on
$\mathfrak{K}$ with coefficients in a set $\Theta$ there is a unique map
$\psi\colon\Theta^{u}\to\Theta$ such that $\theta=\psi\circ\theta^{u}$.

It is clear that the universal signed index is a trait. It appears that some
kind of inverse statement is true.

######  Theorem 1.

Let $\theta^{u}$ be the universal trait with coefficients in a set
$\Theta^{u}$ on a diagram category $\mathfrak{K}$. Then $\theta^{u}$ is the
universal signed index on $\mathfrak{K}$ (with respect to some involution
$\ast$ on the set $\Theta^{u}$).

Let us give a simple but implicit and tautological description of the
universal trait.

######  Definition 10.

Let $\mathfrak{K}$ be a diagram category. The _based diagram category
$\mathfrak{K}_{b}$ associated with $\mathfrak{K}$_ is the category with
objects $(D,v)$ where $D\in\mathfrak{K}$ and $v\in{\mathcal{V}}(D)$ and
morphisms $f\colon(D,v)\to(D^{\prime},v^{\prime})$ such that $f\colon D\to
D^{\prime}$ is a morphism in $\mathfrak{K}$, $v\in dom(f_{*})$ and
$v^{\prime}=f_{*}(v)$.

Informally, the associated based category is category of diagrams with a
distinguished crossing and morphisms (isotopies, isomorphisms and Reidemeister
moves) which don’t erase the distinguished crossing.

Let $\mathfrak{K}$ be a diagram category and $\mathfrak{K}_{b}$ be the
associated based diagram category. Denote the set of objects of the based
category by ${\mathcal{V}}(\mathfrak{K})$. These is an equivalence relation on
the set ${\mathcal{V}}(\mathfrak{K})$: $(D,v)\sim(D^{\prime},v^{\prime})$ iff
there is a morphism $f\colon(D,v)\to(D^{\prime},v^{\prime})$. Denote the set
of equivalence classes by $\overline{{\mathcal{V}}(\mathfrak{K})}$. The
following proposition is evident.

######  Proposition 5.

The map $\theta^{u}$ such that
$\theta^{u}(v)=[(D,v)]\in\overline{{\mathcal{V}}(\mathfrak{K})}$,
$v\in{\mathcal{V}}(D)$, $D\in\mathfrak{K}$, is the universal trait on the
diagram category $\mathfrak{K}$.

Let us give an analogous interpretation of the universal (signed) index.

###  3.2 The graph $\mathcal{G}(\mathfrak{K})$

Let $\mathfrak{K}$ be a diagram category and $\mathfrak{K}_{b}$ be the
associated based diagram category. The set ${\mathcal{V}}(\mathfrak{K})$ can
be identified with the set $\bigsqcup_{D\in\mathfrak{K}}{\mathcal{V}}(D)$ of
crossing of all the diagrams in $\mathfrak{K}$. Let
$\mathcal{G}(\mathfrak{K})$ be the graph with the set of vertex
$V(\mathcal{G}(\mathfrak{K}))={\mathcal{V}}(\mathfrak{K})$ and the set of
edges $E(\mathcal{G}(\mathfrak{K}))$ which consists of edges of two types:

  * •

I0-edge: $vf_{*}(v)$ where $f\colon D\to D^{\prime}$ is an elementary morphism
in $\mathfrak{K}$ and $v$ is a crossing in $dom(f_{*})$;

  * •

I2-edge: $v_{1}v_{2}$ where $v_{1}$ and $v_{2}$ are crossings in a diagram
$D\in\mathfrak{K}$ to which a decreasing second Reidemeister move can be
applied.

Let $\mathcal{G}_{0}(\mathfrak{K})$ be the subgraph in
$\mathcal{G}(\mathfrak{K})$ obtained by removing all I2-edges from
$\mathcal{G}(\mathfrak{K})$.

The proposition below follows from the definitions.

######  Proposition 6.

Let $\mathfrak{K}$ be a diagram category.

  1. 1.

The universal index coefficient set $I^{u}$ is identified with the set of
connected components of the graph $\mathcal{G}(\mathfrak{K})$. The universal
index $\iota^{u}$ maps a crossing to the correspondent connected component.

  2. 2.

The universal trait coefficient set $\Theta^{u}$ is identified with the set of
connected components of the graph $\mathcal{G}_{0}(\mathfrak{K})$. The
universal index $\theta^{u}$ maps a crossing to the correspondent connected
component.

  3. 3.

Let $v\in{\mathcal{V}}(D)$ and $v^{\prime}\in{\mathcal{V}}(D^{\prime})$,
$D,D^{\prime}\in\mathfrak{K}$, be two crossing. Then
$\sigma^{u}(v)=\sigma^{u}(v^{\prime})$ if and only if $v$ and $w$ can be
connected in $\mathcal{G}(\mathfrak{K})$ by a path which contains an even
number of I2-edges, and $\sigma^{u}(v)^{\ast}=\sigma^{u}(v^{\prime})$ if and
only if $v$ and $w$ can be connected by a path which contains an odd number of
I2-edges.

Note that two based diagrams $(D,v)$ and $(D^{\prime},v^{\prime})$ are
equivalent in $\mathfrak{K}_{b}$ iff the crossings $v$ and $v^{\prime}$ belong
to one connected component of the graph $\mathcal{G}_{0}(\mathfrak{K})$.

Thus, if we want to show that the universal trait $\theta^{u}$ coincides with
the universal signed index $\sigma^{u}$ then we have to prove that any path in
$\mathcal{G}(\mathfrak{K})$ which contains an even number of I2-edges, can be
replaced with a path in $\mathcal{G}_{0}(\mathfrak{K})$ with the same ends.
For this, we use a combinatorial trick called _wrapping swap_.

###  3.3 Wrapping

######  Definition 11.

Let $\mathfrak{K}$ be a diagram category and $\mathfrak{K}_{b}$ be the
associated based diagram category. Let $(D,v)$ be a based diagram in
$\mathfrak{K}_{b}$ and $n\in{\mathbb{Z}}$. Define the _wrapped based diagram_
$(D,v,n)$ as the diagram $D^{\prime}$ obtained from $D$ by the rotating the
overcrossing of $v$ counterclockwise by $n$ half-turn as shown in Fig. 18. The
central crossing $v^{\prime}$ of the wrapping in $D^{\prime}$ is the
distinguished crossing of the based diagram $(D,v,n)=(D^{\prime},v^{\prime})$.

Figure 18: Wrappings of a crossing

We denote the wrapped crossing by a disk with the wrapping number $n$ inside.

Note that the original diagram $D$ and the wrapped diagram $D^{\prime}$ are
equivalent in $\mathfrak{K}$.

######  Proposition 7.

Let $\mathfrak{K}$ be a diagram category and $\mathfrak{K}_{b}$ be the
associated based diagram category.

  1. 1.

Let $f\colon(D,v)\to(D^{\prime},v^{\prime})$ is a morphism in
$\mathfrak{K}_{b}$. Then for any $n\in{\mathbb{Z}}$ there is a morphism
$f_{n}\colon(D,v,n)\to(D^{\prime},v^{\prime},n)$ in $\mathfrak{K}_{b}$ of the
wrapped diagrams.

  2. 2.

Let $(D,v)\in\mathfrak{K}_{b}$ be a based diagram. Then for any
$n\in{\mathbb{Z}}$ the wrapped diagrams $(D,v,n)$ and $(D,v,n\pm 2)$ are
equivalent in $\mathfrak{K}_{b}$.

  3. 3.

(wrapping swap) Let $D\in\mathfrak{K}$ be a diagram and
$v_{1},v_{2}\in{\mathcal{V}}(D)$ be two crossings to which a second
Reidemeister move can be applied. Then for any $n\in{\mathbb{Z}}$ the wrapped
diagrams $(D,v_{1},n)$ and $(D,v_{2},n\pm 1)$ are equivalent in
$\mathfrak{K}_{b}$ (Fig. 19).

Figure 19: Wrapping swap

###### Proof.

1\. It is enough to consider an elementary morphism $f$. If $f$ is not a
Reidemeister move or the crossing $v$ does not take part in the move $f$ then
the morphism $f$ induces an elementary morphism
$f_{n}\colon(D,v,n)\to(D^{\prime},v^{\prime},n)$. If $v$ participates in $f$
then $f$ is a third Reidemeister move. In this case we can perform the
analogous move on the wrapped diagrams (see Fig. 20)

Figure 20: Third Reidemeister move with a wrapped crossing

2\. The proof for the second statement is given in Fig. 21. We move one arc
below the wrapped overcrossing, and the other move above the wrapped
overcrossing.

Figure 21: Wrapping reduction

3\. The proof for the wrapping swap is given in Fig. 22. It is realized by one
second Reidemeister move.

Figure 22: Proof for the wrapping swap

∎

######  Remark 6.

If $\mathfrak{K}$ is a diagram category of free tangles then due to
virtualization there can be different types of wrapping (see Fig. 23).

Figure 23: Wrappings with wrapping number $n+1$. The central chord has
wrapping of order $n$

In this case we start proof with the second statement of Proposition 7. There
two types of wrapping of order 2 (up to orientation of tangle components), see
Fig. 24. In the first case we can reduce the wrapping to the single chord $v$
with two second Reidemeister moves, the second case corresponds to the
configuration in Fig. 21 (with $n=0$).

Figure 24: Wrappings of order $2$

Thus, we can consider only wrappings of order $0$ or $1$. Then the other
statements in Proposition 7 are proved by considering possible configurations.

Now we are ready to prove the main theorem.

###  3.4 Proof of the main theorem

By universality of the trait $\theta^{u}$ there is a map
$\psi\colon\Theta^{u}\to S^{u}$ such that $\sigma^{u}=\psi\circ\theta^{u}$. We
will prove that $\psi$ is an injection. Since the universal index and trait
are reduced, this would imply that $\psi$ is a bijection and $\sigma^{u}$ and
$\theta^{u}$ coincide.

Let $v\in{\mathcal{V}}(D)$ and $v^{\prime}\in{\mathcal{V}}(D^{\prime})$,
$D,D^{\prime}\in\mathfrak{K}$, be two crossing such that
$\sigma^{u}(v)=\sigma^{u}(v^{\prime})$. By Proposition 6 that means that there
exists a path $\gamma=v_{1}v_{2}\dots v_{l}$ in $\mathcal{G}(\mathfrak{K})$
such that $v_{1}=v$, $v_{l}=v^{\prime}$ and $\gamma$ contains even number of
I2-edges. Let $D=D_{1},D_{2},\dots,D_{l}=D^{\prime}$ be the diagrams the
crossings $v_{1},\dots,v_{l}$ belong to. We construct a correspondent sequence
of wrapped based diagrams $(D_{i},v_{i},n_{i})$ using induction.

We start with the diagram $(D,v,0)=(D,v)$. Assume we have defined the wrapped
diagram $(D_{i},v_{i},n_{i})$, $1\leq i<l$. If the edge $v_{i}v_{i+1}$ is an
I0-edge then there is an elementary morphism $f\colon D_{i}\to D_{i+1}$ such
that $f_{*}(v_{i})=v_{i+1}$. Then we take $n_{i+1}=n_{i}$, so there is a
morphism $f_{n}\colon(D_{i},v_{i},n_{i})\to(D_{i+1},v_{i+1},n_{i+1})$ in the
based diagram category $\mathfrak{K}_{b}$.

If the edge $v_{i}v_{i+1}$ has type I2 then we set $n_{i+1}=n_{i}\pm 1$ and
consider the wrapping swap
$f_{ws}\colon(D_{i},v_{i},n_{i})\to(D_{i+1},v_{i+1},n_{i+1})$.

Thus, we get a sequence of based diagrams $(D_{i},v_{i},n_{i})$ and morphisms
between them in the based diagram category $\mathfrak{K}_{b}$. That is all
diagrams $(D_{i},v_{i},n_{i})$, $i=1,\dots,l$ are equivalent in
$\mathfrak{K}_{b}$. The sequence finishes with the diagram
$(D_{l},v_{l},n_{l})=(D^{\prime},v^{\prime},n_{l})$. Since only I2-edges
change the wrapping number and the number of such edges in $\gamma$ is even,
the number $n_{l}$ is even. By Proposition 7 the diagram
$(D^{\prime},v^{\prime},n_{l})$ is equivalent to the based diagram
$(D^{\prime},v^{\prime},0)=(D^{\prime},v^{\prime})$.

Thus, the based diagrams $(D,v)$ and $(D^{\prime},v^{\prime})$ are equivalent
in $\mathfrak{K}_{b}$, so by Proposition 5,
$\theta^{u}(v)=\theta^{u}(v^{\prime})$. Theorem 1 is proved.

######  Remark 7.

Since the universal trait and the universal signed index coincide, there is an
involution on the universal trait coefficient set $\Theta^{u}$. In terms of
Proposition 5 we can define it as follows: for any
$[(D,v)]\in\overline{{\mathcal{V}}(\mathfrak{K})}$ its dual element is the
wrapped diagram $[D,v]^{\ast}=[(D,v,1)]$.

###  3.5 Corollaries

Let us formulate several consequences of Theorem 1.

######  Corollary 1.

Let $\mathfrak{K}$ be a diagram category and $\theta$ be a trait on
$\mathfrak{K}$ with coefficients in a set $\Theta$. Then there is a unique map
$\psi\colon S^{u}\to\Theta$ from the coefficient set $S^{u}$ of the universal
signed index $\sigma^{u}$ on $\mathfrak{K}$ such that
$\theta=\phi\circ\sigma^{u}$.

######  Remark 8.

Not all traits are signed indices because the map $\psi$ can be incompatible
with the involution.

For example, let $\mathfrak{K}=\mathfrak{FK}$ be the theory of flat knots, $n$
be the Gaussian signed index and a trait $\theta$ with coefficients in
${\mathbb{Z}}_{2}$ be defined by the formula

$\theta(v)=\left\\{\begin{array}[]{cl}1,&n(v)\in\\{-1,1,2\\},\\\
0,&\mbox{otherwise}.\end{array}\right.$

The trait $\theta$ is neither an index nor a signed index.

######  Theorem 2.

Let $\mathfrak{K}$ be a diagram category. Let $D\in\mathfrak{K}$ and
$v\in{\mathcal{V}}(D)$ be a crossing of the diagram $D$.

  1. 1.

The based diagram $(D,v)$ considered modulo morphisms of $\mathfrak{K}$ which
do not eliminate chosen crossing $v$ is the universal signed index
$\sigma^{u}(v)$ of the crossing $v$.

  2. 2.

The based diagram $(D,v)$ considered modulo morphisms of $\mathfrak{K}$ which
do not eliminate chosen crossing $v$, and wrappings of the chosen crossing, is
the universal index $\iota^{u}(v)$ of the crossing $v$.

######  Remark 9.

For the theory of flat knots $\mathfrak{FK}$, the universal index of a
crossing $v$ in a flat knot diagram $K$ coincides with the equivalence class
$[\tilde{K}^{v}_{glue}]$ of a singular flat knot with one singular point
introduced by A. Henrich [15]. The universal signed index of a crossing $v$ in
a flat knot diagram $K$ coincides with homotopy class $[K_{v}^{+}]$ of signed
singular strings introduced by P. Cahn [1].

##  4 Indistinguishability principle

Now, let us combine the main theorem with the description of the universal
index of tangles in a fixed surface obtained in [40].

Let $D=D_{1}\cup\cdots\cup D_{n}$ be a tangle diagram in a surface $F$. Let
$v\in{\mathcal{V}}(D)$ be a crossing of the diagram $D$.

The _component type_ $\tau(v)$ of the crossing $v$ is the pair $(i,j)$ where
$D_{i}$ is the overcrossing component at $v$ and $D_{j}$ is the undercrossing
component at $v$ (Fig. 25).

Figure 25: A crossing of type $(i,j)$

Let $v$ be a self-crossing of a long component $D_{i}$ and $s_{i}$ be the
starting point of $D_{i}$. Define the _order type_ $o$ of the crossing $v$ to
be $o(v)=-1$ if $v$ is an _early undercrossing_ , i.e. one encounters first
the undercrossing of $v$ while moving along $D_{i}$ from the point $s_{i}$,
and $o(v)=+1$ if $v$ is an _early overcrossing_ , i.e. one encounters the
overcrossing of $v$ first.

The following statements were proved in [40].

######  Theorem 3.

1\. There is no nontrivial indices in the theory of classical knots
$\mathfrak{CK}$.

2\. The order type is the universal index in the theory of long classical
knots $\mathfrak{LCK}$.

3\. The component type $\tau$ is the universal index in the theory of
classical links $\mathfrak{CL}_{n}$ with $n$ (enumerated) components.

Then by Proposition 4 the sign function is the universal signed index on
$\mathfrak{CK}$; the pair $(o,sgn)$ is the universal signed index (with
coefficients in ${\mathbb{Z}}_{2}^{2}$) on $\mathfrak{LCK}$, and $(\tau,sgn)$
is the universal sign index (with coefficients in
$\\{1,\dots,n\\}^{2}\times\\{-1,+1\\}$) on $\mathfrak{CL}_{n}$. By Theorem 1
the above mentioned maps are the universal traits in the corresponding diagram
categories. We can reformulate this fact in the form of indistinguishability
principle.

######  Theorem 4 (Principle of indistinguishability).

  1. 1.

(for classical knots) No crossings of the same sign in a classical knot
diagram can be distinguished by a trait.

  2. 2.

(for long classical knots) No crossings of the same sign and order type in a
classical long knot diagram can be distinguished by a trait.

  3. 3.

(for classical links) No crossings of the same sign and component type in a
classical link diagram can be distinguished by a trait.

Also, we can reformulate the previous theorem as a _substitution principle_. A
manifestation of this principle is shown in Fig. 2.

######  Theorem 5 (Substitution principle for classical knots).

Let $v$ and $w$ be crossing of the same sign in a classical knot diagram $D$.
Then there exist a sequence of Reidemeister moves which transforms $D$ to
itself and moves the crossing $v$ to the crossing $w$.

Analogous statements can be formulated for long classical knots and classical
links.

######  Remark 10.

Analogues of the principles of indistinguishability and substitution can be
formulated for tangles in a fixed surface: if two crossings in a tangle
diagram have the same component, order and homotopy types (see [40]) then one
of them can substitute for the other, and the crossings can not be
distinguished.

######  Remark 11.

There is an analogue of substitution principle for _arcs_ of classical knot
diagrams. By an arc we mean a part of a knot diagram which ends at crossings
and contains no crossings inside.

######  Theorem 6 (Substitution principle for arcs of classical knot diagram).

Let $D$ be a classical knot diagram in the sphere $S^{2}$ and $a,a^{\prime}$
be two arcs of $D$. Then there is a sequence of isotopies and Reidemeiester
moves which transforms $D$ to itself and transforms the arc $a$ to the arc
$a^{\prime}$.

###### Proof.

Choose arbitrary points $A\in a$ and $A^{\prime}\in a^{\prime}$. Rotate the
diagram $D$ on the sphere so that $A$ moves to $A^{\prime}$ and $a$ moves to
an arc tangent to $a^{\prime}$. Denote the diagram after the rotation by
$D^{\prime}$. Choose the point $A^{\prime}$ for the infinity point in
$S^{2}=\mathbb{C}\cup\\{\infty\\}$. The long knot diagrams
$D\setminus\\{A^{\prime}\\}$ and $D^{\prime}\setminus\\{A^{\prime}\\}$ define
one long knot. Thus, there is a sequence of isotopies and Reidemeister moves
in $\mathbb{C}$ which transforms $D^{\prime}$ to $D$. Then the composition of
the initial rotation and this sequence is the required chain of moves. ∎

######  Remark 12.

The analogous statement for _plane_ classical diagrams (diagrams in
${\mathbb{R}}^{2}$) is wrong. One can construct a winding number-like
invariant to distinguish the arcs of plane diagrams.

So we see that the hope to use some inherent information on the crossings for
construction of new invariants of classical knot is false. There are several
ways to deal with this:

  * •

Consider _statistical_ invariants which treat all crossings equally, like
Jones polynomial, Khovanov homology and all the other combinatorial knot
invariants.

  * •

Use not invariant but _equivariant_ labels for crossings. We can require that
the labels of the crossings remain constant up to some isomorphisms. An
example of invariants of this type are parity functors [37] and (bi)quandles
[10]. The substitution principle for arcs does not prevent the existence of
coloring invariants.

  * •

Use nonreidemeister descriptions of knots. There are different decompositions
of knots (into prime knots or JSJ decomposition). May be some of them can be
catched with combinatorial elements of knots other that crossings (for
example, quadrisecants, petal diagrams etc.)

  * •

There still remain virtual knots where indices and parities proved their
worth.

## References

  * [1] P. Cahn, A generalization of Turaev’s virtual string obracket and self-intersections of virtual strings, Communications in Contemporary Mathematics 19:4 (2017) 1650053.
  * [2] J. S. Carter, S. Kamada, and M. Saito, Stable equivalence of knots on surfaces and virtual knot cobordisms J. Knot Theory Ramifications 11:3 (2002) 311–322.
  * [3] Z. Cheng, A polynomial invariant of virtual knots, Proc. Amer. Math. Soc. 142:2 (2014) 713–725.
  * [4] Z. Cheng, A transcendental function invariant of virtual knots, J. Math. Soc. Japan 69:4 (2017) 1583–1599.
  * [5] Z. Cheng, H. Gao and M. Xu, Transcendental linking polynomials of virtual string links, J. Knot Theory Ramifications 28:3 (2019) 1950022.
  * [6] Z. Cheng, The chord index, its definitions, applications and generalizations, Canad. J. Math. 73:3 (2021) 597–621.
  * [7] Z. Cheng, H. Gao and M. Xu, Some remarks on the chord index, J. Knot Theory Ramifications 29:10 (2020) 2042003.
  * [8] Z. Cheng, H. Gao and M. Xu, Chord index for knots in thickened surfaces, arxiv:2109.06439.
  * [9] Z. Cheng, D.A. Fedoseev, H. Gao, V.O. Manturov, M. Xu, From chord parity to chord index, _J. Knot Theory Ramifications_ 29:13 (2020) 2043004.
  * [10] M. Elhamdadi, S. Nelson, _Quandles: An Introduction to the Algebra of Knots_ , AMS, 2015.
  * [11] T. Fiedler, A small state sum for knots, _Topology_ 32:2 (1993) 281–294.
  * [12] L.C. Folwaczny, L.H. Kauffman, A linking number definition of the affine index polynomial and applications, _J. Knot Theory Ramifications_ 22:12 (2013) 1341004.
  * [13] M. Goussarov, M. Polyak, O. Ya. Viro, Finite type invariants of classical and virtual knots, Topology 39:5 (2000) 1045–1068.
  * [14] J. Green, A table of virtual knots, http://www.math.toronto.edu/ drorbn/Students/GreenJ/.
  * [15] A. Henrich, A sequence of degree one Vassiliev invariants for virtual knots, J. Knot Theory Ramifications, 19:4 (2010) 461–487.
  * [16] R. Higa, T. Nakamura, Y. Nakanishi, S. Satoh, The intersection polynomials of a virtual knot, arxiv:2102.12067.
  * [17] D. P. Ilyutko, V. O. Manturov, Cobordisms of Free Knots, Doklady Mathematics 80:3 (2009), 1–3.
  * [18] D. P. Ilyutko, V. O. Manturov, I. M. Nikonov, Virtual Knot Invariants Arising From Parities, Banach Center Publ. 100 (2014) 99–130.
  * [19] D.P. Ilyutko, V.O. Manturov, I.M. Nikonov, Parity and patterns in low-dmensional topology (Cambridge Scietific Publishers, 2015).
  * [20] Y.H. Im, K. Lee, S.Y. Lee, Index polynomial invariant of virtual links, _J. Knot Theory Ramifications_ 19 (2010) 709–725.
  * [21] Y.H. Im, S. Kim, A sequence of polynomial invariants for Gauss diagrams, _J. Knot Theory Ramifications_ 26:7 (2017) 1750039.
  * [22] Y.H. Im, S. Kim, S. Lee, A family of polynomial invariants for knotoids, _J. Knot Theory Ramifications_ 27:11 (2018) 1843001.
  * [23] Y.H. Im, S. Kim, K. Lee, A family of polynomial invariants for flat virtual knots, _Topology Appl._ 264 (2019) 413–419.
  * [24] Y.H. Im, S. Kim, K. Lee, Embeddings from the set of long flat virtual knots to the set of long virtual knots and applications, _J. Knot Theory Ramifications_ 29:5 (2020) 2050025.
  * [25] M. J. Jeong, A zero polynomial of virtual knots, _J. Knot Theory Ramifications_ 25:1 (2016) 1550078.
  * [26] N. Kamada, S. Kamada, Abstract link diagrams and virtual knots, J. Knot Theory Ramifications 9:1 (2000), 93–106.
  * [27] L. H. Kauffman, Virtual Knots, talks at MSRI Meeting in January 1997 and AMS meeting at University of Maryland, College Park in March 1997.
  * [28] L. H. Kauffman, An affine index polynomial invariant of virtual knots, J. Knot Theory Ramifications 22:4 (2013) 1340007.
  * [29] K. Kaur, M. Prabhakar, A. Vesnin, Two-variable polynomial invariants of virtual knots arising from flat virtual knot invariants, J. Knot Theory Ramifications 27:13 (2018) 1842015.
  * [30] G. Kuperberg, What is a Virtual Link?, Alg. Geom. Topol., 3 (2002) 587–591.
  * [31] V. O. Manturov, Parity in Knot Theory, Sb. Math. 201:5-6 (2010) 693–733.
  * [32] V. O. Manturov, Parity, free knots, groups, and invariants of finite type, Trans. Moscow Math. Soc. (2011) 157–169.
  * [33] V. O. Manturov, Parity and cobordisms of free knots, Math. Sb. 203:5 (2012) 196–223.
  * [34] V. O. Manturov, Free knots and parity, in Introductory Lectures on Knot Theory, Selected Lectures Presented at the Advanced School and Conference on Knot Theory and its Applications to Physics and Biology, Series of Knots and Everything 46 (2012) 321–345.
  * [35] V. O. Manturov, Parity and projection from virtual knots to classical knots, J. Knot Theory Ramifications 22:9 (2013) 1350044.
  * [36] I. M. Nikonov, Weak parities and functorial maps, J. Math. Sci. 214:5 (2016) 699–717.
  * [37] I. M. Nikonov, Parity functors, arxiv:2109.12230.
  * [38] I. M. Nikonov, Parity on based matrices, arxiv:2110.04915.
  * [39] I. M. Nikonov, Intersection formulas for parities on virtual knots, arxiv:2110.08392.
  * [40] I. M. Nikonov, Crossing tribes of tangles in a thickened surface, arxiv:2110.12446.
  * [41] V.G. Turaev, Virtual strings, _Ann. Inst. Fourier (Grenoble)_ , 54:7 (2004) 2455–2525.
  * [42] V.G. Turaev, Cobordism of knots on surfaces, _J. Topology_ , 1:2 (2008) 285–305.
  * [43] M.Xu, Writhe polynomial for virtual links, arXiv:1812.05234.

##  Appendix A Examples of indices

###  A.1 Basic indices

######  Example A.1 (The sign).

The sign function is a signed index with coefficients in
${\mathbb{Z}}_{2}=\\{-1,+1\\}$ for the diagram categories of tangles in a
surface and virtual tangles and all their diagram subcategories.

######  Remark A.1.

Often an index can be defined first as a signed index $\sigma$ on flat tangles
and then for virtual tangles, it defines an index $\hat{\sigma}$ using the
sign function (see Proposition 1). This remark can be applied to the following
two basic indices.

######  Example A.2 (Component index).

Let $D=D_{1}\cup\cdots\cup D_{n}$ be a diagram of an oriented flat tangle (or
a flat tangle in a fixed surface), and $v$ be a crossing of $D$. Then $v$ is
an intersection point of components $D_{i}$ and $D_{j}$. We order the
component according to the orientation as shown in Fig. 26 left. The _flat
component (signed) index_ of the crossing $v$ is the ordered pair
$\tau^{f}(v)=(i,j)$. The flat component index $\tau^{f}$ is a signed index
with coefficients in $\\{1,\dots,n\\}^{2}$.

Figure 26: The flat component index (left) and the component index (right)

For a diagram $D=D_{1}\cup\cdots\cup D_{n}$ of an oriented tangle, the
_component index_ $\tau(v)$ of a crossing $v\in{\mathcal{V}}(D)$ is equal to
$(i,j)$ if the component $D_{i}$ overcrosses the component $D_{j}$ at $v$
(Fig. 26 right). It is an index with coefficients in $\\{1,\dots,n\\}^{2}$ on
virtual tangles or tangles in a fixed surface. Note that
$\tau=\hat{\tau}^{f}$.

######  Remark A.2.

On diagram of free tangles one can define a component index with coefficients
in the set of unordered pairs $\\{i,j\\}$, $i,j\in\\{1,\dots,n\\}$.

######  Example A.3 (Order index).

Let $D$ be a diagram of a flat long knot and $v$ be a crossing of $D$. The
oriented smoothing at $v$ splits the diagram into halves $D^{l}_{v}$ and
$D^{r}_{v}$ (Fig 15). Since $D$ is a long knot diagram, one of the halves is
closed. Denote in by $D^{c}_{v}$. We define the _flat order (signed) index_
$o^{f}(v)$ of the crossing $v$ to be equal to $+1$ if $D^{c}_{v}=D^{l}_{v}$
and to be equal to $-1$ if $D^{c}_{v}=D^{r}_{v}$ (Fig. 27). It is a signed
index with coefficients in ${\mathbb{Z}}_{2}=\\{-1,+1\\}$ on flat knot
diagrams.

Figure 27: Flat order index

On diagrams of long virtual knot, one can consider the _order index_
$o=-\hat{o}^{f}$. It is an index with coefficients in ${\mathbb{Z}}_{2}$. The
order index discern the early undercrossings ($o=-1$) and the early
overcrossings ($o=+1$).

######  Remark A.3.

Flat component and order signed indices can play the role of the sign function
for flat knots and tangles in some situations.

###  A.2 Gluing index

The gluing index appeared implicitly in the paper of A. Henrich [15], and a
more general signed gluing index was used by P. Cahn in [1].

######  Definition A.1.

Let $\mathfrak{K}$ be a diagram category of flat tangles (or flat tangles in a
fixed surface). We define the _signed singular diagram category_
$\mathfrak{K}_{ss}$ as follows. The objects of $\mathfrak{K}_{ss}$ are triples
$(D,v,\epsilon)$ where $D\in\mathfrak{K}$, $v\in{\mathcal{V}}(D)$ and
$\epsilon\in\\{-1,+1\\}$. In other words, the diagrams of $\mathfrak{K}_{ss}$
are _signed singular diagrams_ , i.e. flat tangle diagrams with one singular
crossing (“glued” from a flat crossing $v$) marked with a sign $\epsilon=\pm
1$.

The moves on signed flat diagrams are the moves which don’t involve the
singular crossing, and the second and third Reidemeister moves shown in Fig.
28.

Figure 28: Reidemeister moves with the signed singular crossing

Equivalence classes of signed singular diagrams modulo moves on them are
called _signed singular (flat) tangles_.

######  Example A.4 (Gluing index).

Let $\mathcal{S}$ be the set of signed singular flat tangles. There is an
involution on $\mathcal{S}$ which changes the sign of the singular crossing to
the opposite. For a flat tangle diagram $D$ and a crossing
$v\in{\mathcal{V}}(D)$ define its _gluing signed index_
$\sigma_{gl}(v)\in\mathcal{S}$ to be the signed singular flat tangle that
corresponds to the diagram $(D,v,+1)$ with the singular crossing $v$ and the
sign of the crossing $+1$. Then $\sigma_{gl}$ is a signed index on flat tangle
diagrams with coefficients in $\mathcal{S}$. In fact, $\sigma_{gl}$ is the
universal signed index on the flat tangle diagrams.

The linking invariant $lk_{\sigma_{gl}}$ of the signed index is the operator
$\mu$ in [1].

The correspondent index $\bar{\sigma}_{gl}$ takes values in the tangles with
one singular crossing (there is no any signs on the crossing). This index is
defined on diagrams of flat tangles, so on diagrams of virtual tangles. The
linking invariant $lk_{\bar{\sigma}_{gl}}$ is essentially the gluing invariant
$\mathbf{G}$ in [15].

The universality of the index $\sigma_{gl}$ means that other indices on flat
tangles can be presented by compositions $\psi\circ\sigma_{gl}$ where $\psi$
are maps from $\mathcal{S}$ to some sets $I$, i.e. the maps $\psi$ are
$I$-valued invariants of the signed singular flat tangles. Examples of such
indices are given below.

###  A.3 Smoothing index

######  Example 4.

Let $\mathfrak{K}$ be the diagram category of oriented flat tangles (or
oriented flat tangles in a fixed surface $F$) with numbered components. Let
$\mathcal{T}$ be the set of oriented flat tangles (or oriented flat tangles in
$F$), i.e. the set of equivalence classes of diagrams modulo morphisms. Let
$D\in\mathfrak{K}$ be a tangle diagram and $v$ be a crossing of $D$. Consider
the _oriented and unoriented smoothings_ $D^{or}_{v}$ and $D^{un}_{v}$ of the
diagram $D$ at the crossing $v$ (see Fig. 29).

Figure 29: Oriented (left) and unoriented (right) smoothings

The smoothings $D^{or}_{v}$ and $D^{un}_{v}$ are diagrams of oriented tangles
with numbered components. The oriented smoothing has natural orientation. The
orientation of the unoriented smoothing is set forcedly by the orientation
rule in Fig. 29 (the orientation of the lower arc is determined by the
orientation of the upper one).

The _oriented smoothing signed index_ $\sigma_{sm}^{or}(v)$ is the tangle
$[D^{or}_{v}]\in\mathcal{T}$ that corresponds to the oriented smoothing
diagram. It is a signed index with coefficients in $\mathcal{T}$ where the
involution on $\mathcal{T}$ swaps the numbers of the components containing the
left and the right arcs on the smoothed diagram $D^{or}_{v}$ (involution is
trivial if the arcs belong to one component).

The _unoriented smoothing signed index_ $\sigma_{sm}^{un}(v)$ is the tangle
$[D^{un}_{v}]\in\mathcal{T}$ that corresponds to the unoriented smoothing
diagram. It is a signed index with coefficients in $\mathcal{T}$ where the
involution on $\mathcal{T}$ changes the orientation of the smoothed component
in $D^{un}_{v}$.

The index $[\tilde{K}^{d}_{smooth}]$ in [15] and the index $\tilde{L}_{c}$ in
[7] coincide with the index $\bar{\sigma}^{or}_{sm}$ for flat knots. The index
$\tilde{K}_{c}$ in [7] coincides with the index $\bar{\sigma}^{un}_{sm}$ for
flat knots.

Some indices can be deduced from the smoothed indices $\sigma_{sm}^{or}(v)$
and $\sigma_{sm}^{un}(v)$.

######  Example A.5 (Homological index).

Let $\mathfrak{K}$ be the diagram category of a (flat) knot $K$ in a surface
$F$. Let $D\in\mathfrak{K}$ and $v\in{\mathcal{V}}(D)$. The correspondence
$v\mapsto[D^{l}_{v}]$ defines an oriented parity $hp$ with coefficients in
$S=H_{1}(F,{\mathbb{Z}})/[K]$ called the _homological parity_ [18, 37]. Then
$hp=\psi\circ\sigma_{sm}^{or}$ where the map $\psi$ from 2-component links to
$S$ to is defined by the formula $\psi(D_{1}\cup D_{2})=[D_{1}]$.

By definition the homological parity is a signed index, and if $K$ is a non
flat knot we have the index $\widehat{hp}$ such that
$\widehat{hp}(v)=[D^{+}_{v}]\in S$, $v\in{\mathcal{V}}(D)$. The linking
invariant of $\widehat{hp}$ coincides with the small state sum $W_{K}$ in
[11].

The indices $f_{\gamma}$ in [8] are obtained from the homology index using the
map $\psi\colon S\to{\mathbb{Z}}$, $\alpha\mapsto\alpha\cdot\gamma$, where
$\gamma\in H_{1}(F,{\mathbb{Z}})$ and $D\cdot\gamma=0$.

The Gaussian index $n$ can be also deduced from the oriented smoothed index.

######  Example A.6 (Gaussian index).

Let $\mathfrak{K}$ be the diagram category of flat knots. Let
$D\in\mathfrak{K}$ and $v\in{\mathcal{V}}(D)$. The _Gaussian index_ $n(v)$ can
be defined as $n(v)=W(\sigma^{or}_{sm}(v))$ where $W(\cdot)$ is the wriggle
number [12]. The wriggle number $W(D_{1}\cup D_{2})\in{\mathbb{Z}}$ of a two-
component link $D_{1}\cup D_{2}$ can be identified with the linking invariant
$lk_{\tau}(D_{1}\cup D_{2})$ of the component index on the link (the group
$A_{\tau}$ is isomorphic to ${\mathbb{Z}}$ for two-component links).

The Gaussian index $n$ is a signed index with coefficients in ${\mathbb{Z}}$.
For virtual knot diagrams, the corresponding index $\hat{n}=sgn\cdot n$ is the
index $Ind$.

######  Example A.7.

The polynomials $F^{n}_{D}(t,\ell)$ in [29] rely on the combined index
$\iota=(Ind,u_{n}\circ\sigma^{un}_{sm})$ where $u_{n}$ is the coefficient by
the term $t^{n}$ of the (Gaussian) index polynomial for flat knots
$u(t)\in{\mathbb{Z}}[t]$ (see [41]). The index $\iota$ takes values in the set
${\mathbb{Z}}\times{\mathbb{Z}}$.

###  A.4 Based matrix index

Based matrices were defined by V. Turaev in [41]. The based matrix of a knot
diagram carries information about intersection numbers for halves of the
diagram at the crossings. So based matrices can be considered as homological
analogues of knot diagrams. Then an analogue of the gluing signed index
$\sigma_{gl}$ are singed singular based matrices defined in [1].

######  Definition A.2.

A _signed singular based matrix_ is a quadruple $T=(G,s,d,b,\epsilon)$ where
$G$ is a finite set, $s,d\in G$, $s\neq d$, $b\colon G\times G\to{\mathbb{Z}}$
is a skew-symmetric map and $\epsilon\in\\{-1,+1\\}$.

An element $g\in G\setminus\\{s\\}$ is _annihilating_ if $b(g,h)=0$ for any
$h\in G$.

An element $g\in G\setminus\\{s\\}$ is _core_ if $b(g,h)=b(g,s)$ for any $h\in
G$.

Two elements $g_{1},g_{2}\in G\setminus\\{s\\}$ are _complementary_ if
$b(g_{1},h)+b(g_{2},h)=b(g,s)$ for any $h\in G$.

We consider the following _elementary extensions_ :

  * $M1$

Given a signed singular based matrix $T=(G,s,d,b,\epsilon)$, form the signed
singular based matrix $\bar{T}=(\bar{G},s,d,\bar{b},\epsilon)$ such that
$\bar{G}=G\sqcup\\{g\\}$, $\bar{b}|_{G\times G}=b$ and $\bar{b}(g,h)=0$ for
all $h\in G$.

  * $M2$

Given $T=(G,s,d,b,\epsilon)$, form the signed singular based matrix
$\bar{T}=(\bar{G},s,d,\bar{b},\epsilon)$ such that $\bar{G}=G\sqcup\\{g\\}$,
$\bar{b}|_{G\times G}=b$ and $\bar{b}(g,h)=b(s,h)$ for all $h\in G$.

  * $M3$

Given $T=(G,s,d,b,\epsilon)$, form the signed singular based matrix
$\bar{T}=(\bar{G},s,d,\bar{b},\epsilon)$ such that
$\bar{G}=G\sqcup\\{g_{1},g_{2}\\}$, $\bar{b}|_{G\times G}=b$, and $g_{1}$ and
$g_{2}$ are complementary in $\bar{T}$, i.e.
$\bar{b}(g_{1},h)+\bar{b}(g_{2},h)=b(s,h)$ for all $h\in G$.

  * $N$

Given $T=(G,s,d,b,\epsilon)$ and an element $g\in G\setminus\\{s\\}$ such that
$d$ and $g$ are complementary, form the signed singular based matrix
$\bar{T}=(G,s,g,b,-\epsilon)$.

A signed singular based matrix $T$ is called _primitive_ if the inverse
excisions $M1^{-1},M2^{-1},M3^{-1}$ can not be applied to $T$.

Two signed singular based matrix $T$ and $T^{\prime}$ are _homologous_ if
$T^{\prime}$ can be obtained from $T$ by a sequence of elementary excisions
$M1,M2,M3,N$ and the inverse transformations to them.

The model example for a signed singular based matrix is the following. Let
$(D,v,\epsilon)$ be a signed singular diagram of a flat knot. Then the
quintuple $T(D,v,\epsilon)=(G,s,d,b,\epsilon)$ where
$G=\\{s\\}\sqcup{\mathcal{V}}(D)$, $d=v$,
$b(w,w^{\prime})=[D^{l}_{w}]\cdot[D^{l}_{w^{\prime}}]$,
$w,w^{\prime}\in{\mathcal{V}}(D)$, is the intersection number of the knot
halves, and $b(s,w)=[D]\cdot[D^{l}_{w}]$, $w\in{\mathcal{V}}(D)$, is a signed
singular base matrix.

With the correspondence $(D,v,\epsilon)\mapsto T(D,v,\epsilon)$, Reidemeister
moves on signed singular diagrams turn into moves $M1,M2,M3,N$.

######  Proposition A.1 ([1]).

1\. If two signed singular diagrams $(D,v,\epsilon)$ and
$(D^{\prime},v^{\prime},\epsilon^{\prime})$ are equivalent then the signed
singular based matrices $T(D,v,\epsilon)$ and
$T(D^{\prime},v^{\prime},\epsilon^{\prime})$ are homologous.

2\. For any signed singular based matrix $T$ there exist a primitive signed
singular based matrix $T_{\bullet}=(G_{\bullet},s,d,b_{\bullet},\epsilon)$
which is homologous to $T$. The based matrix $T_{\bullet}$ is unique up to
isomorphisms, the move $N$, and the replacement of the core element $d$ by an
annihilating element together with simultaneous change of $\epsilon$ by
$-\epsilon$ (in the case when $d$ is a core element).

Thus, $T_{\bullet}(D,v,\epsilon)$ is an invariant of signed singular based
matrices (up to isomorpisms and the moves described in Proposition A.1).

######  Example A.8 (Based matrix index).

Let $\mathfrak{FK}$ be the diagram category of flat knots. Let
$D\in\mathfrak{FK}$ and $v\in{\mathcal{V}}(D)$. The _based matrix signed
index_ $\sigma_{bm}(v)$ of the crossing $v$ is defined by the formula
$\sigma_{bm}(v)=T(\sigma_{gl}(v))=T_{\bullet}(\sigma_{gl}(v))\in\mathcal{M}$
where $\mathcal{M}$ is the set of homology classes of signed singular based
matrices ($\mathcal{M}$ is also equal to the set of primitive signed singular
based matrices up to moves described in Proposition A.1). The involution on
$\mathcal{M}$ changes the sign $\epsilon$ of a signed singular based matrix to
the opposite.

The correspondent index $\bar{\sigma}_{bm}(v)$ appears in [15].

######  Remark A.4.

The reduced stable parity on based matrices in [38] is obtained by applying
the intersection formula [39] to an index which can be deduced from the index
$\bar{\sigma}_{bm}(v)$.

For virtual knots, the definition of based matrices can be refined by
involving the sign function [42]. A _graded signed singular based matrix_ is a
quadruple $(G,s,d,b,sgn)$ where $G$ is a set, $s,d\in G$, $b\colon G\times
G\to{\mathbb{Z}}$ is a skew-symmetric form and $sgn$ is a function from
$G\setminus\\{s\\}$ to $\\{-1,+1\\}$. The sign of the singular based matrix is
$\epsilon=sgn(d)$.

The refined definition of complementary elements are defined as follows:
$g_{1},g_{2}\in G\setminus\\{s\\}$ are _complementary_ if
$sgn(g_{1})=-sgn(g_{2})$ and $b(g_{1},h)+b(g_{2},h)=b(s,h)$ for all $h\in G$.
The refined elementary excisions $M3$ and $N$ use the new definition of
complementarity.

Given a diagram $D$ of a virtual knot and a crossing $v\in{\mathcal{V}}(D)$,
the definition of the signed singular based matrix
$T(D,v,sgn(v))=(G,s,d,b,sgn(v))$ extends to the graded signed singular based
matrix $T_{gr}(D,v)=(G,s,d,b,sgn)$.

With the sign function in a graded signed singular based matrix
$T=(G,s,d,b,sgn)$, one can define the forms $b^{\alpha\beta}\colon G\times
G\to{\mathbb{Z}}$, $\alpha,\beta\in\\{-1,+1\\}$, by the formulas

$\displaystyle b^{\alpha\beta}(g,h)=\alpha\beta
sgn(g)sgn(h)b(g,h)-\frac{1-\alpha sgn(g)}{2}b(g,s)-\frac{1-\beta
sgn(h)}{2}b(s,h),$ $\displaystyle b^{\alpha\beta}(g,s)=\alpha
sgn(g)b(g,s),\quad b^{\alpha\beta}(s,h)=\beta sgn(h)b(s,h),\quad
b^{\alpha\beta}(s,s)=0,$

for $g,h\in G\setminus\\{s\\}$. For complementary elements $g_{1},g_{2}\in G$
of $T$ we have the relation
$b^{\alpha\beta}(g_{1},h)=b^{\alpha\beta}(g_{2},h)$ for any $h\in G$ and any
$\alpha,\beta\in\\{-1,+1\\}$. Hence, the
$\iota^{\alpha\beta}(g)=b^{\alpha\beta}(d,g)$ is an index with coefficients in
${\mathbb{Z}}$ on the category of graded signed singular based matrices with
excision moves. Let
$p^{\alpha\beta}(T)=lk_{\iota^{\alpha\beta}}(T)\in{\mathbb{Z}}[{\mathbb{Z}}]$
be the corresponding linking invariant.

######  Example A.9 (Intersection index).

Let $\mathfrak{VK}$ be the virtual knot theory. Let $D\in\mathfrak{VK}$ be a
diagram and $v$ be a crossing in $D$. The _intersection index_
$i^{\alpha\beta}(v)$ of the crossing $v$ is defined as
$i^{\alpha\beta}(v)=p^{\alpha\beta}\circ
T_{gr}(D,v)\in{\mathbb{Z}}[{\mathbb{Z}}]$. The the linking invariants of the
indices $i^{\alpha\beta}$, $\alpha,\beta\in\\{-1,+1\\}$ can be identified (up
to terms with the loop index values) with the polynomials $f_{ij}$ in [16].

###  A.5 Secondary index

Let us recall the definition of the oriented parity [36, 37].

######  Definition A.3.

Let $\mathfrak{K}$ be a diagram category. An _oriented parity_ $p$ on
$\mathfrak{K}$ is a signed index with coefficients in an abelian group $A$
(with the involution $x^{*}=-x$, $x\in A$) which obeys the following
conditions:

  * (P0)

if a first Reidemeister move can be applied to a crossing $v$ in a diagram
$D\in\mathfrak{K}$ then $p(v)=0$.

  * (P3+)

if $f\colon D\to D^{\prime}$ is a third Reidemeister move then

$\epsilon_{\Delta}(v_{1})\cdot p_{D}(v_{1})+\epsilon_{\Delta}(v_{2})\cdot
p_{D}(v_{2})+\epsilon_{\Delta}(v_{3})\cdot p_{D}(v_{3})=0$

where $v_{1},v_{2},v_{3}$ are the crossings involved in the move and
$\epsilon_{\Delta}(v_{i})$ is the incidence index of the crossing $v_{i}$ to
the disappearing triangle $\Delta$, see Fig. 30.

Figure 30: Incidence indices

An examples of oriented parities are the homological parity (Example A.5) and
the Gaussian index $n$ (Example A.6).

######  Example A.10 (Secondary index).

Let $p$ be an oriented parity with coefficients in an abelian group $A$ on the
diagram category $\mathfrak{FK}$ of flat knots. Let $\tilde{A}=\bigoplus_{a\in
A}{\mathbb{Z}}[A/\langle a\rangle]$, and $\ast$ be the involution on
$\tilde{A}$ such that $(A_{a})^{\ast}=A_{-a}$, $a\in A$, and

$\left(\sum_{x\in A/\langle
a\rangle}\lambda_{x}\cdot[x]\right)^{\ast}=-\sum_{x\in A/\langle
a\rangle}\lambda_{x}\cdot[-x],\quad a\in A.$

For a diagram $D\in\mathfrak{VK}$ and a crossing $v\in{\mathcal{V}}(D)$ define
the _secondary index_ of the crossing $v$ by the formula

$\sigma_{p}(v)=\sum_{v^{\prime}\in{\mathcal{V}}(D)}lk(v,v^{\prime})\cdot[lk(v,v^{\prime})\cdot
p(v^{\prime})]\in{\mathbb{Z}}[A/\langle p(v)\rangle]\subset\tilde{A}$

where the number $lk(v,v^{\prime})$ is defined a shown in Fig. 31 (the diagram
$D$ is considered as a Gauss diagram; the chords in flat knot diagrams are
oriented as if they were positive crossings). We set $lk(v,v)=0$ for any
$v\in{\mathcal{V}}(D)$.

Figure 31: The number $lk(v,v^{\prime})$

######  Proposition A.2.

The map $\sigma_{p}$ is a signed index on $\mathfrak{FK}$ with coefficients in
$\tilde{A}$.

###### Proof.

Let us check the property (I0). Let $D\in\mathfrak{FK}$,
$v\in{\mathcal{V}}(D)$ and $f\colon D\to D$ be an increasing second
Reidemeister move with new crossings
$w_{1},w_{2}\in{\mathcal{V}}(D^{\prime})$. Then $p(w_{1})=-p(w_{2})$ and
$lk(f_{*}(v),w_{1})=-lk(f_{*}(v),w_{2})$. Then

$\sigma_{p}(f_{*}(v))=\sum_{w^{\prime}\in{\mathcal{V}}(D^{\prime})}lk(f_{*}(v),w^{\prime})\cdot[lk(f_{*}(v),w^{\prime})\cdot
p(w^{\prime})]=\\\
\sum_{w\in{\mathcal{V}}(D)}lk(f_{*}(v),f_{*}(w))\cdot[lk(f_{*}(v),f_{*}(w))\cdot
p(f_{*}(w))]+\\\ lk(f_{*}(v),w_{1})\cdot[lk(f_{*}(v),w_{1})\cdot
p(w_{1})]+lk(f_{*}(v),w_{2})\cdot[lk(f_{*}(v),w_{2})\cdot p(w_{2})]=\\\
\sum_{w\in{\mathcal{V}}(D)}lk(v,w)\cdot[lk(v,w)\cdot p(w)]=\sigma_{p}(v).$

Let $f\colon D\to D^{\prime}$ be a third Reidemeister move shown in Fig. 32.
Then $lk(v,w)=1$, $lk(u^{\prime},v^{\prime})=-1$ and
$lk(u^{\prime},w^{\prime})=1$. By the property (P3+) $p(u)-p(v)-p(w)=0$.

Figure 32: The number $lk(v,v^{\prime})$

Then

$\displaystyle\sigma_{p}(f_{*}(u))-\sigma_{p}(u)=-[-p(v^{\prime})]+[p(w^{\prime})]=0,$
$\displaystyle\sigma_{p}(f_{*}(v))-\sigma_{p}(v)=[p(u^{\prime})]-[p(w)]=0,$
$\displaystyle\sigma_{p}(f_{*}(u))-\sigma_{p}(u)=-[-p(u^{\prime})]+[-p(v)]=0,$

because $-p(u^{\prime})=-p(u)\equiv p(v)\mod p(w)$ etc. The other cases of
third Reidemeister moves are proved analogously.

Let us check the property (I2+). Let $v_{1},v_{2}\in{\mathcal{V}}(D)$,
$D\in\mathfrak{FK}$, be crossings to which a second Reidemeister move can be
applied. Then $p(v_{1})=-p(v_{2})\equiv 0\mod p(v_{i})$, $i=1,2$;
$lk(v_{1},v_{2})=-lk(v_{2},v_{1})$, and $lk(v_{1},v)=-lk(v_{2},v)$ for any
$v\in{\mathcal{V}}(D)\setminus\\{v_{1},v_{2}\\}$. Thus,
$\sigma_{p}(v_{2})=\sigma_{p}(v_{1})^{\ast}$. ∎

The transcendental index polynomials in [4, 5, 8] rely on the secondary index
$\iota_{p}=\hat{\sigma}_{p}$ for some oriented parities $p$.

###  A.6 Derived parities

A series of signed indices (in fact, oriented parities) on virtual knot
diagrams was constructed in [39].

######  Example A.11 (Derived parities).

Let $\mathfrak{K}=\mathfrak{K}(K)$ be the diagram category of a virtual knot
$K$, and $p$ be an oriented parity on $\mathfrak{K}$ with coefficients in an
abelian group $A$. Then the _derived parity_ $p^{\prime}$ on the diagram
category $\mathfrak{K}$ is given by the formula

$p^{\prime}(v)=\sum_{v^{\prime}\in{\mathcal{V}}(D)}p(v^{\prime})\cdot(D^{l}_{v}\cdot
D^{-}_{v^{\prime}})\in A^{\prime},\quad
v\in{\mathcal{V}}(D),D\in\mathfrak{K},$

where $A^{\prime}=A/\langle\sum_{v\in{\mathcal{V}}(D)}p(v)\cdot
Ind(v)\rangle$. By definition $p^{\prime}$ is a signed index with coefficients
in $A^{\prime}$.

One can define also the higher derivatives $p^{\prime\prime}$,
$p^{\prime\prime\prime}$ and so on.

###  A.7 Indices via parity projection

Recall the definition of a weak parity [31].

######  Definition A.4.

Let $\mathfrak{K}$ be a diagram category. A _weak parity_ $\psi$ on the
diagram category $\mathfrak{K}$ is an index with coefficients in
${\mathbb{Z}}_{2}$ which satisfies the condition

  * $(\Psi 3)$

if $f\colon D\to D^{\prime}$, $D,D^{\prime}\in\mathfrak{K}$, is a third
Reidemeister move on crossings $u,v,w\in{\mathcal{V}}(D)$ and
$\psi(u)=\psi(v)=0$ then $\psi(w)=0$.

Any weak parity $\psi$ induces a natural transformation $\Psi$ (called the
_parity projection_ induced by $\psi$) from the diagram category
$\mathfrak{K}$ to the diagram category of virtual tangles (category of
flat/free tangles if $\mathfrak{K}$ consists of diagrams of flat/free tangles)
as follows: given a diagram $D\in\mathfrak{K}$, the diagram $\Psi(D)$ is
obtained by replacing all classical crossings $v\in{\mathcal{V}}(D)$ such that
$\psi(v)=1$ by virtual crossing. The parity projection $\Psi$ defines a
correct map from the knots, links or tangles of $\mathfrak{K}$ (i.e. the
equivalence classes of diagrams to $\mathfrak{K}$) to virtual (or flat or
free) tangles.

######  Proposition A.3 ([36]).

Let $p$ be an ordered parity on $\mathfrak{K}$ with coefficients in an abelian
group $A$. Then the map $\psi_{p}$ given by the formula

$\psi_{p}(v)=\left\\{\begin{array}[]{cc}1,&p(v)\neq 0,\\\
0,&p(v)=0,\end{array}\right.$

$v\in{\mathcal{V}}(D)$, $D\in\mathfrak{K}$, is weak parity on $\mathfrak{K}$.

######  Example A.12 (Index induced by a parity projection).

Let $\mathfrak{D}$ be a diagram category, $\psi$ be a weak parity on
$\mathfrak{K}$, $\Psi$ be the correspondent parity projection to the diagram
category $\mathfrak{VT}$ of virtual tangles, and $\iota$ be an index on
$\mathfrak{VT}$ with coefficients in a set $I$. The the _index induced from
$\iota$ by the parity projection $\Psi$_ is the index $\psi(\iota)$ on
$\mathfrak{K}$ with coefficients in $I\sqcup\\{\bullet\\}$ defined by the
formula

$\psi(\iota)(v)=\left\\{\begin{array}[]{cc}\bullet,&\psi(v)\neq 0,\\\
\iota(\Psi_{*}(v)),&p(v)=0,\end{array}\right.$

$v\in{\mathcal{V}}(D)$, $D\in\mathfrak{K}$.

The index polynomials in [21, 22, 23, 24, 25] use the indices induced from the
Gaussian index $Ind$ by the weak parity $\psi_{Ind_{n}}$ where
$Ind_{n}=Ind\mod n$, $n\in{\mathbb{N}}\cup\\{0\\}$ is the Gaussian index
considered modulo $n$.

###  A.8 Biquandle index

Recall that a _biquandle_ [10] is a set $B$ with two operations
$\ast,\circ\colon B\times B\to B$ which satisfy the following conditions:

  1. 1.

$x\circ x=x\ast x$ for all $x\in B$;

  2. 2.

the maps $(x,y)\mapsto(y,x\circ y)$, $(x,y)\mapsto(x,y\ast x)$ and
$(x,y)\mapsto(x\circ y,y\ast x)$ are bijections of $B\times B$

  3. 3.

for any $x,y,z\in B$;

$\displaystyle(x\circ y)\circ(z\circ y)=(x\circ z)\circ(y\ast z),$
$\displaystyle(x\circ y)\ast(z\circ y)=(x\ast z)\circ(y\ast z),$
$\displaystyle(x\ast y)\ast(z\ast y)=(x\ast z)\ast(y\circ z).$

For a diagram $D$ of a virtual tangle, a _colouring_ of $D$ with a biquandle
$B$ is a map $c$ from the set of the arcs in $D$ to $B$ which obeys the
colouring rule in Fig. 33 for any crossing of $D$.

Figure 33: The colouring rule

Let $Col_{B}(D)$ denote the set of colourings of the diagram $D$. For any
diagram $D^{\prime}$ of the same tangle, there is a bijection between the
colouring sets $Col_{B}(D)$ and $Col_{B}(D^{\prime})$.

Let us describe the (signed) index introduced in [6].

######  Example A.13 (Biquandle index).

Let $B$ be a biquandle. Consider the set
$\tilde{B}=\tilde{B}^{-}\sqcup\tilde{B}^{+}$ where $B^{\pm}$ are the quotient
sets

$\displaystyle\tilde{B}^{+}=B/\langle(x,y)=(x\circ z,y\circ z)=(x\ast z,y\ast
z)=(x\circ z,y\ast z),\ x,y,z\in B\rangle,$
$\displaystyle\tilde{B}^{-}=B/\langle(x,y)=(x\circ z,y\circ z)=(x\ast z,y\ast
z)=(x\ast z,y\circ z),\ x,y,z\in B\rangle,$

with the involution $\ast\colon\tilde{B}^{\pm}\to\tilde{B}^{\mp}$,
$(x,y)^{\ast}=(y,x)$, $x,y\in B$.

Let $\mathfrak{K}$ be a diagram category of virtual tangles. Let
$D\in\mathfrak{K}$ and $Col_{B}(D)$ be the set of colourings of $D$. For any
colouring $c\in Col_{B}(D)$ and any crossing $v\in{\mathcal{V}}(D)$ consider
the element $\sigma_{B,c}(v)=(x,y)\in\tilde{B}^{sgn(v)}$ where $x,y$ are the
colours of arcs incident to $v$ (see Fig. 33). The _biquandle signed index_
$\sigma_{B}(v)$ of the crossing $v$ is the multiset
$\sigma_{B}(v)=\\{\sigma_{D,c}(v)\\}_{c\in Col_{B}(D)}$.

From the biquandle properties, the following statement holds.

######  Proposition A.4.

The map $\sigma_{B}$ is a signed index on $\mathfrak{K}$ with coefficients in
the multi-subsets of $\tilde{B}$.

######  Remark A.5.

The map $\sigma_{B,c}$ can be considered as a _local signed index_. It
satisfies the property (I2+) and the following property: for any morphism
$f\colon D\to D^{\prime}$ and any crossing $v\in dom(f_{*})$ we have
$\sigma_{B,f_{*}(c)}(f_{*}(v))=\sigma_{B,c}(v)$ where $f_{*}(c)\in
Col_{B}(D^{\prime})$ is the colouring which corresponds to the colouring $c$.
Note that $\sigma_{B,c}$ may not define a (global) signed index due to the
colouring monodromy.

######  Example A.14.

Let $B={\mathbb{Z}}$ be the biquandle with operations $x\circ y=x\ast y=x+1$.
Then $\tilde{B}^{\pm}\simeq{\mathbb{Z}}$ by the isomorphism $(x,y)\mapsto
x-y$. Then the index $\sigma_{B,c}$ coincides with the Gaussian signed index
$n$ on the virtual knots.

###  A.9 An example

Consider the knot 3.1 from Green’s table [14]. Let us calculate some of the
indices described above.

Figure 34: The knot 3.1

  1. 1.

The gluing signed index $\sigma_{gl}$ values of the crossings are shown in
Fig. 35

Figure 35: The gluing index $\sigma_{gl}$ on the knot 3.1

  2. 2.

The values of the smoothing signed indices $\sigma_{sm}^{or}$ and
$\sigma_{sm}^{un}$ on the crossings are shown in Fig. 36

Figure 36: The smoothing signed indices $\sigma_{sm}^{or}$ (top) and
$\sigma_{sm}^{un}$ (bottom) on the knot 3.1

  3. 3.

The values of the based matrix sign index $\sigma_{bm}$ on the crossings are

$\left(\begin{array}[]{r|rrr}0&\mathbf{-1}&-1&2\\\
\hline\cr\mathbf{1}&\mathbf{0}&\mathbf{0}&\mathbf{2}\\\ 1&\mathbf{0}&0&1\\\
-2&\mathbf{-2}&-2&0\end{array}\right)^{-}\
\left(\begin{array}[]{r|rrr}0&-1&\mathbf{-1}&2\\\ \hline\cr
1&0&\mathbf{0}&2\\\ \mathbf{1}&\mathbf{0}&\mathbf{0}&\mathbf{1}\\\
-2&-2&\mathbf{-2}&0\end{array}\right)^{+}\
\left(\begin{array}[]{r|rrr}0&-1&-1&\mathbf{2}\\\ \hline\cr
1&0&0&\mathbf{2}\\\ 1&0&0&\mathbf{1}\\\
\mathbf{-2}&\mathbf{-2}&\mathbf{-2}&\mathbf{0}\end{array}\right)^{-}$

The matrix is the matrix of the form $b$; the first column and row correspond
to the element $s$; the column and the row which correspond to $d$ are
highlighted with the bold font; the sign $\epsilon$ is the matrix superscript.

  4. 4.

The Gaussian index is $n(1)=-1$, $n(2)=-1$, $n(3)=2$.

  5. 5.

The secondary index of the Gaussian index is trivial for the first and the
second crossings, and takes the value
$\sigma_{n}(3)=2\cdot[1]\in{\mathbb{Z}}[{\mathbb{Z}}_{2}]$ for the third
crossing.

  6. 6.

The derived parity $n^{\prime}$ of the Gaussian index has coefficients in
${\mathbb{Z}}_{4}$ and is equal to $n^{\prime}(1)=-1$, $n^{\prime}(2)=1$,
$n^{\prime}(3)=-1$. The second derivative $n^{\prime\prime}$ has coefficients
in ${\mathbb{Z}}_{4}$ and is equal to $n^{\prime\prime}(1)=-1$,
$n^{\prime\prime}(2)=0$, $n^{\prime\prime}(3)=1$.

  7. 7.

Consider the weak parity $\psi=\psi_{Ind_{2}}$. Then the induced index
$\iota=\psi(Ind)$ is equal $\iota(1)=\iota(2)=\bullet$, $\iota(3)=0$.

  8. 8.

Let $B={\mathbb{Z}}$ be the biquandle with operations $x\circ y=x\ast y=z+1$.
Then by Remark A.5 the biquandle signed index $\sigma_{B}$ is isomorphic to
the Gaussian index $n$ and takes the values $-1,-1,2$ on the crossings of the
knot 3.1.