# Symmetric diagrams for all strongly invertible knots up to 10 crossings

Christoph Lamm

###### Abstract.

We present a table of symmetric diagrams for strongly invertible knots up to
10 crossings, point out the similarity of transvergent diagrams for strongly
invertible knots with symmetric union diagrams and discuss open questions.

###### Key words and phrases:

strongly invertible knots

###### 2020 Mathematics Subject Classification:

57K10

##  1\. Introduction

A knot $K\subset\mathbb{S}^{3}$ is said to be strongly invertible if there is
an orientation preserving smooth involution $h$ of $\mathbb{S}^{3}$ such that
$h(K)=K$ and $h$ reverses the orientation on $K$.

Sakuma’s article ‘On strongly invertible knots’ [11] which appeared in 1986
contains a table of symmetric diagrams of strongly invertible knot up to
crossing number 9. Strongly invertible knots currently receive a lot of
attention, see for instance [1], [2], [9] and [12]. Our own motivation to
extend the table of symmetric diagrams to knots with crossing number 10 stems
from the related symmetry type of symmetric unions of knots and a project to
study strongly positive amphicheiral knots.

Strongly invertible knots can be depicted as transvergent diagrams (in which
the axis of rotation lies in the diagram plane) or as intravergent diagrams
(where the axis is perpendicular to the diagram plane). The purpose of this
article is to give one transvergent diagram for each (prime) strongly
invertible knot with $c(K)\leq 10$.

######  Definition 1.1.

For a knot $K$ we denote, as usual, by $c(K)$ the minimal crossing number of a
diagram of $K$. If $K$ is strongly invertible we denote by $c_{t}(K)$ the
minimal crossing number of all transvergent diagrams of $K$.

Symmetric equivalence can be defined for strongly invertible knots either
using the conjugacy class of $h$ in the symmetry group of $K$ or by symmetric
diagram moves (see [9], Section 2). Interestingly, especially when compared to
the case of symmetric unions [3], there are always one or two equivalent
classes (two equivalent classes occuring for knots with period 2). Whereas
Sakuma lists diagrams showing both classes simultaneously (for knots having
two of them), in this article we contend with only one diagram for each knot.
We hope to extend this in a later version to a complete list of diagrams for
knots with two equivalence classes.

We focus on the crossing number of the symmetric diagrams and try to find
minimal (transvergent) diagrams. In many cases symmetric diagrams exist with
$c(K)$ crossings, so that $c_{t}(K)=c(K)$. These symmetric diagrams with
minimal crossing number are occasionally also contained in knot tables (for
instance in Rolfsen’s table the knots $9_{39}$, $9_{40}$, $9_{41}$ are shown
with a vertical symmetry axis). For our list we cannot guarantee that the
crossing number of a diagram is minimal if it exceeds $c(K)$ because we did
not exhaustively check all symmetric diagrams.

##  2\. Motivation

Symmetric unions of knots were introduced by Kinoshita and Terasaka in 1957
[5]. They consist of the connected sum of a knot and its mirror image with
reversed orientation, with crossings inserted on the symmetry axis. The
building blocks of transvergent diagrams of strongly invertible knots are very
similar: A connected sum of a knot diagram and a rotational copy, with
crossings inserted on the symmetry axis. Figure 1 shows an example for this
similarity. In a future study we will compare methods for symmetric unions and
strongly invertible knots and check if the transfer of results is possible.

Figure 1. The knot $9_{46}$. Left: as a symmetric union (mirror symmetry),
right: as a strongly invertible diagram (rotational symmetry with inversion of
orientation).

Another aim is the construction of strongly positive amphicheiral knots as
symmetric unions with strongly invertible partial knots. The diagrams used for
this construction possess two symmetry axes and a rotation around an axis
perpendicular to the diagram plane yields the strongly positive amphicheiral
property, see Figure 2.

Figure 2. A strongly positive amphicheiral knot diagram with two axes of
symmetry (showing the knot 12a1019)

A knot is called strongly positive amphicheiral if it has a diagram which is
mapped to its mirror image by a rotation of $\pi$, preserving the orientation.
Obviously, composite knots consisting of a knot and its mirror image are
strongly positive amphicheiral. Prime strongly positive amphicheiral knots are
rare, however: until recently only three prime strongly positive amphicheiral
knots with 12 or fewer crossings were known: $10_{99}$, $10_{123}$ and 12a427.
In [8] we found additional examples, so that currently the list consists of
$10_{99}$, $10_{123}$, 12a427, 12a1019, 12a1105, 12a1202, and 12n706. A future
study will extend this list to strongly positive amphicheiral knots with
crossing numbers up to 16.

##  3\. Template notation and generation of the table

All two-bridge knots are strongly invertible and because they are also
2-periodic they have two equivalent classes of strong invertibility, see [11].
Sakuma shows how to find representatives for each knot and class. Therefore we
do not need to list diagrams of two-bridge knots in detail and we concentrate
on the remaining knots up to crossing number 10 (which all have bridge number
3). For two-bridge knots we will discuss the question whether symmetric
diagrams exist with $c(K)$ crossings, however.

Our tabulating approach uses templates, in a similar way as in [8]. As half of
a diagram contains already the information for the whole diagram, a template
only shows the upper half and integer markings on the axis for the twists
inserted on the axis. The axis is placed horizontally.

Figure 3 shows how a template diagram is constructed from a transvergent
diagram.

Figure 3. Constructing a template representation from a transvergent diagram
(the example shows $B_{1}(-2,-1,-1)=10_{76}$)

In the other direction we proceed as in Figure 4: If a template diagram is
given we need to reconstruct the other half and insert the crossings on the
axis.

Figure 4. Converting a template into a knot diagram (the example shows
$C_{1}(1,-2,1)=10_{122}$)

The list of all strongly invertible 3-bridge knots in the Appendix was
compiled in the following way: We started with Sakuma’s table and converted
his diagrams into templates. Variation of the twist attributes on the axis
then yielded many 10 crossing knots (using Knotscape [7]). Other sources, as
Rolfsen’s table and examples in [2] and [9] were used in the same way. The
remaining 8 cases were manually transformed into symmetric diagrams with the
help of the software KLO [6], see the Acknowledgments. From the collection of
all examples we chose one diagram with smallest crossing number for our table.
As we remarked, it is expected that transvergent diagrams exist with smaller
crossing number for some of them.

##  4\. Two-bridge knots

For two-bridge knots we use the Conway notation $C(a_{1},\ldots,a_{n})$ (and a
shorter version $[a_{1},\ldots,a_{n}]$ in Figure 7). As usual, the plat
diagram’s closing patterns are different for even and odd $n$, see Figures 5
and 6.

Figure 5. Horizontal symmetry for alternating two-bridge knot diagrams Figure
6. Vertical symmetry for alternating two-bridge knot diagrams

In these figures we give examples for $n=6$ (horizontal placement and axis)
and $n=5$ (vertical case, but still with horizontal axis).

Sakuma’s representatives for the two equivalence classes use ‘even’ continued
fractions. Each two-bridge knot can be written as $C(a_{1},\ldots,a_{n})$ with
even numbers $a_{i}$. In this case $n$ is also even and half of $n$ is the
knot’s genus. If all $a_{i}$ are even then the flyping modification shown in
Figure 5 is possible. However, to proceed as in the figure, only ever second
half-twist number needs be even.

In order to find diagrams with $c_{t}(K)=c(K)$ we therefore try to transform
the Conway notation of each 2-bridge knot so that it realizes the minimal
crossing number and has even entries for ever second $a_{i}$. For example, for
2-bridge torus knots we may use $T(2,2n+1)=C(2n+1)=C(1,2n)$. This condition is
the first part of the following proposition. The second part uses a symmetric
diagram which is placed vertically and is illustrated in Figure 6.

######  Proposition 4.1.

A two-bridge knot $K$ has a transvergent diagram with $c(K)$ crossings, if

  * •

$K$ can be written as $C(a_{1},\ldots,a_{n})$ with even $n$ and all $a_{i}>0$
where the twist-numbers $a_{2}$, $a_{4}$, …are even, or

  * •

$K$ can be written as $C(a_{1},\ldots,a_{n})$ with odd $n$ and all $a_{i}>0$
and symmetric twist-numbers $a_{1}=a_{n}$, $a_{2}=a_{n-1}$, …. In this case
$a_{\frac{n+1}{2}}$ is necessarily odd.

######  Conjecture 4.2.

A two-bridge knot $K$ has a transvergent diagram with $c(K)$ crossings if and
only if the above conditions are fulfilled.

For crossing numbers less than 8 we illustrate this in Figure 7. Note, that
for the knot $6_{3}=C(2,1,1,2)$ neither of the two conditions of Proposition
4.1 is realized so that we conjecture that no transvergent diagram with 6
crossings exists for this knot.111This should be easy to show, enumerating all
alternating diagrams with 6 crossings. Because of the Tait conjecture for
alternating knots the proof of Conjecture 4.2 shouldn’t be too difficult
either. An extended project would then try to characterize 2-bridge knots with
$c_{t}(K)=c(K)+1$.

Figure 7. Two-bridge knots with $c\leq 7$: Symmetric diagrams with minimal
crossing number

There are five 2-bridge knots with 8 crossings which do not satisfy one of the
two conditions: $8_{7}=C(4,1,1,2)$, $8_{8}=C(2,3,1,2)$, $8_{9}=C(3,1,1,3)$,
$8_{13}=C(3,1,1,1,2)$, $8_{14}=C(2,2,1,1,2)$.

##  5\. Three-bridge knots

The result of the generation of transvergent diagrams as described in Section
3 is shown in Table 1. The template names require some explanation: We use
$B$, $C$, $D$ and $E$ to denote template families. Family $B$ collects curves
which can be simplified by Reidemeister 1 moves. (The even simpler Family $A$
is used for the standard representations of two-bridge knots and is not yet
depicted.) Curves in Family $C$ contain a trefoil or, as in case $C_{4}$, a
related diagram with a trefoil shadow. Family $D$ features the figure eight
knot as half-diagram and Family $E$ the shadow of the torus knot $T(2,5)$. In
all three cases of Family $E$ the curve is a trefoil because the crossings are
not alternating.

8 a | $8_{5}$ | $B_{1}$ | 8 | 10 a | $10_{46}$ | $B_{4}$ | 10 | 10 a | $10_{77}$ | $B_{1}$ | 11 | 10 n | $10_{132}$ | $C_{3a}$ | 11
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| $8_{10}$ | $B_{1}$ | 9 |  | $10_{47}$ | $B_{4}$ | 11 |  | $10_{78}$ | $C_{3a}$ | 10 |  | $10_{133}$ | $D_{1}$ | 11
| $8_{15}$ | $C_{1}$ | 8 |  | $10_{48}$ | $B_{3}$ | 11 |  | $10_{89}$ | $D_{1}$ | 10 |  | $10_{134}$ | $B_{3}$ | 11
| $8_{16}$ | $C_{1}$ | 8 |  | $10_{49}$ | $B_{3}$ | 12 |  | $10_{96}$ | $E_{1}$ | 13 |  | $10_{135}$ | $B_{3}$ | 12
| $8_{18}$ | $C_{1}$ | 8 |  | $10_{50}$ | $B_{1}$ | 10 |  | $10_{97}$ | $E_{2}$ | 13 |  | $10_{136}$ | $C_{1}$ | 10
|  |  |  |  | $10_{51}$ | $B_{1}$ | 11 |  | $10_{99}$ | $D_{1}$ | 10 |  | $10_{137}$ | $B_{2}$ | 10
8 n | $8_{19}$ | $C_{1}$ | 8 |  | $10_{52}$ | $B_{1}$ | 11 |  | $10_{100}$ | $C_{3a}$ | 10 |  | $10_{138}$ | $D_{1}$ | 10
| $8_{20}$ | $B_{2}$ | 8 |  | $10_{53}$ | $B_{1}$ | 12 |  | $10_{101}$ | $C_{3a}$ | 11 |  | $10_{139}$ | $C_{4}$ | 10
| $8_{21}$ | $C_{1}$ | 8 |  | $10_{54}$ | $B_{4}$ | 11 |  | $10_{103}$ | $C_{1}$ | 10 |  | $10_{140}$ | $B_{5}$ | 10
|  |  |  |  | $10_{55}$ | $B_{4}$ | 12 |  | $10_{104}$ | $C_{2a}$ | 10 |  | $10_{141}$ | $C_{3b}$ | 10
9 a | $9_{16}$ | $B_{1}$ | 9 |  | $10_{56}$ | $B_{4}$ | 12 |  | $10_{105}$ | $C_{2a}$ | 11 |  | $10_{142}$ | $C_{1}$ | 10
| $9_{22}$ | $B_{1}$ | 10 |  | $10_{57}$ | $B_{4}$ | 13 |  | $10_{108}$ | $C_{1}$ | 10 |  | $10_{143}$ | $B_{5}$ | 11
| $9_{24}$ | $B_{1}$ | 10 |  | $10_{58}$ | $B_{1}$ | 10 |  | $10_{111}$ | $C_{1}$ | 10 |  | $10_{144}$ | $C_{1}$ | 10
| $9_{25}$ | $B_{1}$ | 10 |  | $10_{59}$ | $B_{1}$ | 11 |  | $10_{112}$ | $C_{3a}$ | 10 |  | $10_{145}$ | $C_{2a}$ | 11
| $9_{28}$ | $C_{6}$ | 10 |  | $10_{60}$ | $D_{1}$ | 10 |  | $10_{113}$ | $C_{3a}$ | 11 |  | $10_{146}$ | $C_{3a}$ | 11
| $9_{29}$ | $E_{2}$ | 12 |  | $10_{61}$ | $C_{1}$ | 10 |  | $10_{114}$ | $C_{1}$ | 10 |  | $10_{152}$ | $E_{3}$ | 11
| $9_{30}$ | $B_{1}$ | 11 |  | $10_{62}$ | $B_{5}$ | 11 |  | $10_{116}$ | $D_{1}$ | 10 |  | $10_{154}$ | $E_{3}$ | 11
| $9_{34}$ | $C_{1}$ | 9 |  | $10_{63}$ | $C_{1}$ | 10 |  | $10_{120}$ | $C_{1}$ | 10 |  | $10_{155}$ | $C_{3b}$ | 10
| $9_{35}$ | $C_{1}$ | 9 |  | $10_{64}$ | $C_{2a}$ | 10 |  | $10_{121}$ | $C_{1}$ | 10 |  | $10_{156}$ | $C_{2a}$ | 11
| $9_{36}$ | $B_{1}$ | 9 |  | $10_{65}$ | $B_{5}$ | 12 |  | $10_{122}$ | $C_{1}$ | 10 |  | $10_{157}$ | $C_{2b}$ | 10
| $9_{37}$ | $C_{1}$ | 9 |  | $10_{66}$ | $C_{3a}$ | 10 |  | $10_{123}$ | $D_{1}$ | 10 |  | $10_{158}$ | $C_{1}$ | 10
| $9_{38}$ | $E_{1}$ | 12 |  | $10_{68}$ | $C_{2a}$ | 11 |  |  |  |  |  | $10_{159}$ | $C_{3b}$ | 10
| $9_{39}$ | $C_{1}$ | 9 |  | $10_{69}$ | $C_{3a}$ | 11 | 10 n | $10_{124}$ | $B_{6}$ | 10 |  | $10_{160}$ | $C_{1}$ | 10
| $9_{40}$ | $C_{1}$ | 9 |  | $10_{70}$ | $B_{1}$ | 10 |  | $10_{125}$ | $B_{3}$ | 11 |  | $10_{161}$ | $C_{4}$ | 10
| $9_{41}$ | $C_{1}$ | 9 |  | $10_{71}$ | $B_{1}$ | 11 |  | $10_{126}$ | $B_{3}$ | 10 |  | $10_{162}$ | $C_{1}$ | 10
|  |  |  |  | $10_{72}$ | $B_{1}$ | 11 |  | $10_{127}$ | $B_{3}$ | 11 |  | $10_{163}$ | $C_{1}$ | 10
9 n | $9_{42}$ | $C_{1}$ | 9 |  | $10_{73}$ | $B_{1}$ | 12 |  | $10_{128}$ | $C_{1}$ | 10 |  | $10_{164}$ | $C_{1}$ | 10
| $9_{43}$ | $B_{2}$ | 9 |  | $10_{74}$ | $C_{5}$ | 10 |  | $10_{129}$ | $B_{2}$ | 10 |  | $10_{165}$ | $C_{1}$ | 10
| $9_{44}$ | $B_{2}$ | 9 |  | $10_{75}$ | $C_{6}$ | 10 |  | $10_{130}$ | $B_{2}$ | 10 |  |  |  |
| $9_{45}$ | $C_{1}$ | 10 |  | $10_{76}$ | $C_{2a}$ | 10 |  | $10_{131}$ | $B_{2}$ | 11 |  |  |  |
| $9_{46}$ | $C_{1}$ | 9 |  |  |  |  |  |  |  |  |  |  |  |
| $9_{47}$ | $C_{1}$ | 9 |  |  |  |  |  |  |  |  |  |  |  |
| $9_{48}$ | $C_{1}$ | 9 |  |  |  |  |  |  |  |  |  |  |  |
| $9_{49}$ | $C_{1}$ | 9 |  |  |  |  |  |  |  |  |  |  |  |
Table 1. A list of all 118 prime strongly invertible knots with bridge number
3. The second column indicates the template where the diagram information is
found and the third column contains the crossing number of the diagram.

Note, that some templates generate many diagrams and other just a few. A
discussion of some of these cases will be inserted here in a later version.

As remarked before, we chose one diagram for each knot for this list. For
instance, the knot $10_{76}$ is assigned to template $C_{2a}$ with a symmetric
diagram with 10 crossings, $10_{76}=C_{2a}(0,0,-2)$. We could also have
assigned it to template $B_{1}$, because, as shown in Figure 3,
$10_{76}=B_{1}(-2,-1,-1)$ in a 10 crossing diagram.

##  6\. A comparison of symmetries

In this section we will compare several involutive symmetries of knots, see
Figures 8 and 9. For a diagram $D$ we denote by $D^{r}$ the rotated diagram,
by $-D$ the mirrored diagram (at a plane perpendicular to the diagram plane)
with reversed orientation, and by $D^{*}$ the diagram mirrored in the diagram
plane.

Figure 8. Three symmetries: Intravergent diagrams are shown on the left and
transvergent diagrams on the right. Symmetric unions are symmetric to a plane
of reflection and all diagrams are transvergent. Figure 9. Two amphicheiral
symmetries: Strongly positive and negative amphicheiral knots are point
symmetric and all diagrams are intravergent.

##  7\. Comparison of ribbons spanning symmetric unions and strongly
invertible knots

As a first topic in the comparison of methods for symmetric unions and
strongly invertible knots we look at the symmetric ribbon which spans a
symmetric union [8]. Because one half in a corresponding diagram for a
strongly invertible knot differs from the symmetric union just by switching
crossings we obtain a similar ribbon but with clasp singularities, see Figure
10. The clasp number of a knot $K$ is the minimum number of clasp
singularities among all disks spanning $K$. The article [4] contains a table
of the clasp numbers of knots up to crossing number 10 (some of the values are
not yet known).

Example: For $10_{142}$, shown in Figure 10 on the left, we find in [4] that
the clasp number is 3 and the diagram contains 3 clasp singularities. For
another knot, $10_{139}$, the clasp number is 4 and it is listed in our table
with template $C_{4}$ which generates 4 clasp singularities in the ribbon
(because there are 4 crossings in the template). We conclude that this knot
cannot be generated with templates $B_{1}$, $B_{2}$ or $C_{3}$.

Therefore, transferring the use of the ribbon from the symmetric union case to
the strongly invertible case may give some interesting results and, as
mentioned, we propose to do that in more detail in a future article.

Figure 10. A comparison of the singularities in a band (a) left: spanning a
strongly invertible knot, $10_{142}=C_{1}(0,4,0)$, with clasp singularities
and (b) right: a symmetric union, $10_{140}$, with ribbon singularities.

##  8\. Plausibility check for alternating diagrams and related examples

We distinguish alternating and non-alternating templates – defined in the same
way as for knot diagrams. Only alternating templates can generate transvergent
diagrams for alternating knots with $c(K)$ crossings. On the other hand, non-
alternating templates may generate transvergent diagrams for alternating
knots, which then have necessarily more than $c(K)$ crossings.

###  8.1. Alternating diagrams

Table 1 and the Appendix contain 8 alternating and 10 non-alternating
templates. For the 8 alternating templates we list in Table 2 the alternating
knots with transvergent diagrams having $c(K)$ crossings. For each template
there is a specific condition on the twist numbers: the diagram generated from
the template is alternating if and only if this condition is satisfied. This
is a useful plausibility check for the computational results.

$B_{1}$ |  $t_{1}\leq 0$, $t_{2}\leq 0$, $t_{3}\leq 0$ |  $8_{5}$, $9_{16}$, $9_{36}$, $10_{50}$, $10_{58}$, $10_{70}$
---|---|---
$B_{4}$ |  $t_{1}\leq 0$, $t_{2}\leq 0$, $t_{3}\leq 0$ | $10_{46}$
$C_{1}$ |  $t_{1}\geq 0$, $t_{2}\leq 0$, $t_{3}\geq 0$ |  all 18 alternating knots listed with template $C_{1}$
$C_{2a}$ |  $t_{1}\geq 0$, $t_{2}\leq 0$, $t_{3}\leq 0$ |  $10_{64}$, $10_{76}$, $10_{104}$
$C_{3a}$ |  $t_{1}\geq 0$, $t_{2}\leq 0$, $t_{3}\geq 0$, $t_{4}\geq 0$ |  $10_{66}$, $10_{78}$, $10_{100}$, $10_{112}$
$C_{5}$ |  $t_{1}\leq 0$, $t_{2}\leq 0$ | $10_{74}$
$C_{6}$ |  $t_{1}\geq 0$, $t_{2}\geq 0$ | $10_{75}$
$D_{1}$ |  $t_{1}\geq 0$, $t_{2}\geq 0$, $t_{3}\leq 0$, $t_{4}\leq 0$ |  $10_{60}$, $10_{89}$, $10_{99}$, $10_{116}$, $10_{123}$
Table 2. The alternating templates, the conditions on the twist numbers for
alternating diagrams and the respective alternating knots

######  Remark 8.1.

There are alternating knots for which Table 1 contains non-alternating
diagrams and these obviously do not satisfy the conditions in Table 2. For
instance $10_{72}=B_{1}(-1,-1,3)$ is an alternating knot and listed with
template $B_{1}$ as a diagram with 11 crossings. The twist numbers $t_{1}=-1$
and $t_{2}=-1$ satisfy the condition $t_{1}\leq 0$, $t_{2}\leq 0$ but the
third twist number does not, since $t_{3}>0$. The similar cases $10_{47}$,
$10_{54}$ and $10_{55}$ are illustrated in the next section in Figure 11.

###  8.2. Alternating knots with ‘nearly alternating diagrams’

Table 1 and the Appendix contain some diagrams for alternating knots with
$c(K)+1$ crossings which can be transformed by a ‘flip’ to alternating
diagrams with $c(K)$ crossings, see Figure 11 (and there is even the case
$10_{55}$ with two flips, transforming a symmetric diagram with 12 crossings
to a minimal alternating one).

Since these flips are especially simple, we list the cases in Table 3. In each
of these cases a twist number of 2 occurs in a template where in the last
section we noted the condition $t_{i}\leq 0$, or a twist number of -2 where we
noted the condition $t_{i}\geq 0$. This corresponds to the fact that the
diagrams are not alternating inside the areas marked in gray.

$B_{1}$ |  $8_{10}$, $9_{24}$, $9_{25}$, $9_{30}$, $10_{51}$, $10_{53}$, $10_{71}$, $10_{77}$
---|---
$B_{4}$ |  $10_{47}$, $10_{54}$, $10_{55}$
$C_{2a}$ |  $10_{68}$, $10_{105}$
$C_{3a}$ |  $10_{69}$, $10_{101}$, $10_{113}$
Table 3. Symmetric diagrams in alternating templates which can be transformed
by ‘flips’ to alternating diagrams Figure 11. In each case flipping the gray
area(s) yields an alternating diagram: $10_{47}=B_{4}(-1,0,2)$,
$10_{54}=B_{4}(2,0,-1)$ and $10_{55}=B_{4}(2,0,2)$

######  Remark 8.2.

Similar ‘nearly minimal’ diagrams exist for non-alternating knots, for
instance $10_{124}$, $10_{134}$ in the non-alternating template $B_{3}$, and
$10_{146}$ in the alternating template $C_{3a}$.

##  9\. Open questions and suggested projects

The knot invariant $c_{t}(K)$ is currently only known for knots for which
transvergent diagrams with $c(K)$ crossings were experimentally found, and in
this case $c_{t}(K)=c(K)$. This situation can be improved in computational or
theoretical ways.

######  Project 9.1.

Generate all transvergent diagrams up to $b$ crossings and determine their
knot types. Since for 3-bridge knots our list contains diagrams with $\leq 13$
crossings, it is enough to set $b=12$ in order to obtain $c_{t}(K)$ for all
(strongly invertible, prime) 3-bridge knots up to crossing number 10.

######  Project 9.2.

Find a knot invariant which allows the computation of a lower bound for
$c_{t}(K)$ for each (strongly invertible) knot $K$.

On the other hand, we would like to know more about the difference
$c_{t}(K)-c(K)$:

######  Project 9.3.

Find an upper bound for $d(n):=\max_{K}\\{c_{t}(K)-c(K)|c(K)=n\\}$, for
strongly invertible knots $K$. It might be reasonable to study $d(n)$
separately for 2-bridge knots and knots with a larger bridge number.

## Acknowledgments

I thank Marc Kegel for his contribution of 8 symmetric diagrams in the
Mathoverflow discussion ‘Diagrams for strongly invertible knots with 10
crossings’ [10] which I started in May 2022.

##  10\. Appendix

The Appendix contains the 18 templates generating 118 diagrams of prime
strongly invertible knots with bridge number 3 and crossing number $\leq 10$.
The twist numbers correspond to the dotted lines in the templates read from
left to right.

Family B

template $B_{1}$

$8_{5\phantom{0}}$ | = | ( 0, | -1, | -1)
---|---|---|---|---
$8_{10}$ | = | ( 0, | -1, | 2)
$9_{16}$ | = | (-1, | -1, | -1)
$9_{22}$ | = | ( 0, | -1, | 3)
$9_{24}$ | = | (-1, | -1, | 2)
$9_{25}$ | = | ( 0, | -2, | 2)

$9_{30}$ | = | ( 0, | 2, | 3)
---|---|---|---|---
$9_{36}$ | = | ( 0, | -2, | -1)
$10_{50}$ | = | ( 0, | -3, | -1)
$10_{51}$ | = | ( 0, | -3, | 2)
$10_{52}$ | = | ( 0, | -1, | 4)
$10_{53}$ | = | ( 0, | 2, | 4)
$10_{58}$ | = | ( 0, | -2, | -2)
$10_{59}$ | = | ( 0, | -2, | 3)
$10_{70}$ | = | (-1, | -2, | -1)
$10_{71}$ | = | (-1, | -2, | 2)
$10_{72}$ | = | (-1, | -1, | 3)
$10_{73}$ | = | (-1, | 2, | 3)
$10_{77}$ | = | (-2, | -1, | 2)

template $B_{2}$

$\phantom{1}8_{20}$ | = | (-1, | 1)
---|---|---|---
$\phantom{1}9_{43}$ | = | (-1, | 2)
$\phantom{1}9_{44}$ | = | (-2, | 1)
$10_{129}$ | = | (-3, | 1)
$10_{130}$ | = | (-1, | 3)
$10_{131}$ | = | ( 2, | 3)
$10_{137}$ | = | (-2, | 2)

template $B_{3}$

$10_{48}$ | = | (-1, | 1, | -1)
---|---|---|---|---
$10_{49}$ | = | (-1, | 1, | 2)
$10_{125}$ | = | ( 1, | 1, | -1)
$10_{126}$ | = | ( 0, | 1, | -1)
$10_{127}$ | = | ( 0, | 1, | 2)
$10_{134}$ | = | ( 0, | -2, | -1)
$10_{135}$ | = | ( 0, | -2, | 2)

template $B_{4}$

$10_{46}$ | = | (-1, | 0, | -1)
---|---|---|---|---
$10_{47}$ | = | (-1, | 0, | 2)
$10_{54}$ | = | ( 2, | 0, | -1)
$10_{55}$ | = | ( 2, | 0, | 2)
$10_{56}$ | = | (-1, | 2, | -1)
$10_{57}$ | = | (-1, | 2, | 2)

template $B_{5}$

$10_{62}$ | = | (1, | 1, | -1)
---|---|---|---|---
$10_{65}$ | = | (1, | -2, | -1)
$10_{140}$ | = | (1, | 0, | -1)
$10_{143}$ | = | (1, | -1, | -1)

template $B_{6}$

$10_{124}$ | = | ( 1, | 1)
---|---|---|---

Family C

template $C_{1}$

$\phantom{1}8_{15}$ | = | ( 0, | 0, | 2)
---|---|---|---|---
$\phantom{1}8_{16}$ | = | ( 0, | -1, | 1)
$\phantom{1}8_{18}$ | = | ( 1, | 0, | 1)
$\phantom{1}8_{19}$ | = | ( 0, | 2, | 0)
$\phantom{1}8_{21}$ | = | (-2, | 0, | 0)
$\phantom{1}9_{34}$ | = | ( 1, | 0, | 2)
$\phantom{1}9_{35}$ | = | ( 0, | -3, | 0)
$\phantom{1}9_{37}$ | = | ( 0, | 0, | 3)
$\phantom{1}9_{39}$ | = | ( 0, | -1, | 2)
$\phantom{1}9_{40}$ | = | ( 1, | -1, | 1)
$\phantom{1}9_{41}$ | = | ( 0, | -2, | 1)
$\phantom{1}9_{42}$ | = | ( 0, | 2, | 1)
$\phantom{1}9_{45}$ | = | ( 1, | 1, | 2)
$\phantom{1}9_{46}$ | = | ( 0, | 3, | 0)

$\phantom{1}9_{47}$ | = | (-2, | 0, | 1)
---|---|---|---|---
$\phantom{1}9_{48}$ | = | (-3, | 0, | 0)
$\phantom{1}9_{49}$ | = | (-2, | -1, | 0)
$10_{61}$ | = | ( 0, | -4, | 0)
$10_{63}$ | = | ( 0, | 0, | 4)
$10_{103}$ | = | ( 0, | -1, | 3)
$10_{108}$ | = | ( 0, | -3, | 1)
$10_{111}$ | = | ( 0, | -2, | 2)
$10_{114}$ | = | ( 1, | 0, | 3)
$10_{120}$ | = | ( 2, | 0, | 2)
$10_{121}$ | = | ( 1, | -1, | 2)
$10_{122}$ | = | ( 1, | -2, | 1)
$10_{128}$ | = | ( 0, | 2, | 2)
$10_{136}$ | = | ( 1, | 2, | 1)
$10_{142}$ | = | ( 0, | 4, | 0)
$10_{144}$ | = | (-4, | 0, | 0)
$10_{158}$ | = | (-3, | -1, | 0)
$10_{160}$ | = | ( 0, | 3, | 1)
$10_{162}$ | = | (-2, | -2, | 0)
$10_{163}$ | = | (-3, | 0, | 1)
$10_{164}$ | = | (-2, | -1, | 1)
$10_{165}$ | = | (-2, | 0, | 2)

template $C_{2a}$

$10_{64}$ | = | ( 0, | -1, | -1)
---|---|---|---|---
$10_{68}$ | = | ( 0, | -1, | 2)
$10_{76}$ | = | ( 0, | 0, | -2)
$10_{104}$ | = | ( 1, | 0, | -1)
$10_{105}$ | = | ( 1, | 0, | 2)
$10_{145}$ | = | ( 0, | 2, | -1)
$10_{156}$ | = | ( 1, | 1, | -1)

template $C_{2b}$

$10_{157}$ | = | ( 1, | 0, | 1)
---|---|---|---|---

template $C_{3a}$

$10_{66}$ | = | ( 0, | 0, | 1, | 1)
---|---|---|---|---|---
$10_{69}$ | = | ( 0, | 0, | 1, | -2)
$10_{78}$ | = | ( 0, | 0, | 0, | 2)
$10_{100}$ | = | ( 0, | -1, | 0, | 1)
$10_{101}$ | = | ( 0, | -1, | 0, | -2)
$10_{112}$ | = | ( 1, | 0, | 0, | 1)
$10_{113}$ | = | ( 1, | 0, | 0, | -2)
$10_{132}$ | = | ( 0, | -1, | -1, | 1)
$10_{146}$ | = | ( 0, | 0, | -2, | 1)

template $C_{3b}$

$10_{141}$ | = | ( 0, | 0, | -1, | -1)
---|---|---|---|---|---
$10_{155}$ | = | ( 0, | -1, | 0, | -1)
$10_{159}$ | = | ( 1, | 0, | 0, | -1)

template $C_{4}$

$10_{139}$ | = | ( 1, | 1)
---|---|---|---
$10_{161}$ | = | (-1, | 1)

template $C_{5}$

$10_{74}$ | = | (-1, | -1)
---|---|---|---

template $C_{6}$

$9_{28}$ | = | (-2, | 0)
---|---|---|---
$10_{75}$ | = | ( 1, | 1)

Family D

template $D_{1}$

$10_{60}$ | = | ( 2, | 0, | 0, | 0)
---|---|---|---|---|---
$10_{89}$ | = | ( 1, | 0, | 0, | -1)
$10_{99}$ | = | ( 0, | 1, | 0, | -1)
$10_{116}$ | = | ( 1, | 1, | 0, | 0)
$10_{123}$ | = | ( 1, | 0, | -1, | 0)
$10_{133}$ | = | ( 1, | 1, | 0, | 1)
$10_{138}$ | = | ( 0, | 0, | 2, | 0)

Family E

template $E_{1}$

$9_{38}$ | = | (-1, | 1)
---|---|---|---
$10_{96}$ | = | (-1, | 2)

template $E_{2}$

$9_{29}$ | = | ( 1, | 1)
---|---|---|---
$10_{97}$ | = | ( 2, | 1)

template $E_{3}$

$10_{152}$ | = | ( 1)
---|---|---
$10_{154}$ | = | (-1)

## References

  * [1] K. Boyle: Involutions of alternating links, Proc. Amer. Math. Soc. 149 (2021), 3113–3128
  * [2] K. Boyle and A. Issa: Equivariant 4-genera of strongly invertible and periodic knots, Journal of Topology 15 (2022), 1635–1674
  * [3] M. Eisermann and C. Lamm: Equivalence of symmetric union diagrams, J. Knot Theory Ramifications 16 (2007), 879–898
  * [4] T. Kadokami and K. Kawamura: An infinite family of prime knots with a certain property for the clasp number, J. Knot Theory Ramifications 23, 1450071 (2014)
  * [5] S. Kinoshita and H. Terasaka: On unions of knots, Osaka Math. J. 9 (1957), 131–153
  * [6] F. Swenton: Knot-Like Objects (KLO, software, version 0.977 alpha)
  * [7] J. Hoste and M. Thistlethwaite: Knotscape – providing convenient access to tables of knots (software), 1999
  * [8] C. Lamm: The search for nonsymmetric ribbon knots, Experimental Mathematics 30 (2021), 349–363
  * [9] A. Lobb and L. Watson: A refinement of Khovanov homology, Geom. Topol. 25 (2021), 1861–1917
  * [10] MathOverflow, Question ‘Diagrams for strongly invertible knots with 10 crossings’, May 22, 2022
  * [11] M. Sakuma: On strongly invertible knots, Algebraic and Topological Theories – to the memory of Dr. Takehiko Miyata (1986), 176–196
  * [12] M. Snape: Homological invariants of strongly invertible knots, Ph.D. thesis, University of Glasgow, 2018

Christoph Lamm
Rückertstr. 3, 65187 Wiesbaden
Germany
e-mail<EMAIL_ADDRESS>


#  YITP-22-98
Real tensor eigenvalue/vector distributions
of the Gaussian tensor model via a four-fermi theory

Naoki<EMAIL_ADDRESS>
Yukawa Institute for Theoretical Physics, Kyoto University,
and
CGPQI, Yukawa Institute for Theoretical Physics, Kyoto University,
Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan

Eigenvalue distributions are important dynamical quantities in matrix models,
and it is an interesting challenge to study corresponding quantities in tensor
models. We study real tensor eigenvalue/vector distributions for real
symmetric order-three random tensors with the Gaussian distribution as the
simplest case. We first rewrite this problem as the computation of a partition
function of a four-fermi theory. The partition function is exactly computed
for some small-$N$ cases, and is shown to precisely agree with Monte Carlo
simulations. For large-$N$, it seems difficult to compute it exactly, and we
apply an approximation using a self-consistency equation for two-point
functions and obtain an analytic expression. It turns out that the real tensor
eigenvalue distribution is simply the Gaussian within this approximation. We
compare the approximate expression with Monte Carlo simulations, and find
that, if an extra overall factor depending on $N$ is multiplied to the the
expression, it agrees well with the Monte Carlo results. It is left for future
study to improve the approximation for large-$N$ to correctly derive the
overall factor.

##  1 Introduction

Eigenvalue distributions are important dynamical quantities in matrix models.
The distributions give quantitative/qualitative insights into complex
dynamical systems with strongly coupled degrees of freedom, such as atomic
systems [1]. Topological transitions of the distributions [2, 3] give
important insights into the properties of gauge theories and two-dimensional
quantum gravity, such as phase structures in particular. The distributions are
also important as one of the main tools to solve the matrix models [4].

Considering recent attentions to tensor models [5, 6, 7, 8] it would be
interesting to study corresponding quantities and their roles in tensor
models. While, in various applications [9]222See for example [10] for the
tensor model perspectives on such applications., it is important to develop
techniques to obtain eigenvalues/vectors [11, 12, 13] for given tensors, it is
rather more important to understand statistical properties of tensor
eigenvalues/vectors for ensembles of tensors in tensor models, where tensors
are dynamical. In fact, not many results are known about their statistical
properties: The expectation values of the numbers of real eigenvalues are
computed for random real tensors [14, 15]; the sizes of the largest
eigenvalues are estimated for random tensors [16]; Wigner semi-circle law in
matrix models has been extended to tensor models [17].

A basic question about eigenvalues/vectors in tensor models is their
distributions. Rather surprisingly, as far as we are aware, no systematic
results had not been given until recently, when the current author derived an
exact formula333The formula is expressed with the confluent hypergeometric
function of the second kind. for signed distributions of real eigenvectors for
real symmetric order-three random tensors with the Gaussian distribution [18]:
Each eigenvector contributes to the distribution by $\pm 1$ depending on the
sign of an associated Hessian matrix. An interesting intermediate step was
that the problem was rewritten as the computation of a partition function of a
four-fermi theory. This was achieved by Gaussian integrations over bosonic
variables after rewriting the problem as a partition function of a system of
bosonic and fermionic variables. The final formula was obtained by exactly
computing the partition function of the four-fermi theory.

The above procedure of the previous paper can be extended to other kinds of
eigenvalue/vector distributions. The purpose of the present paper is to extend
it to the distributions of real eigenvalues/vectors for real symmetric order-
three random tensors with the Gaussian distribution. In the previous paper, we
only dealt with one couple of fermions, but, in this paper, we deal with $R$
replicas of fermions and take analytic continuation to $R=1/2$. We derive a
four-fermi theory with the replicated fermions, which is more complex than
that of the previous paper. We exactly compute the partition function of the
four-fermi theory for some small-$N$ cases and show the precise agreement with
Monte Carlo results. On the other hand, for large-$N$, we apply an
approximation using the Schwinger-Dyson equation (See Appendix A). We find
that the obtained approximate analytic expression has good agreement with
Monte Carlo results, if we multiply an extra overall factor depending on $N$
to the expression, where $N$ denotes the range of the tensor indices.

This paper is organized as follows. In Section 2, we define the distribution
of the real tensor eigenvalues/vectors of the Gaussian tensor model, and
rewrite the distribution as a partition function of a system of bosonic and
fermionic variables. In Section 3, by Gaussian integrations over the bosonic
variables of the system, we obtain a four-fermi theory. In Section 4, we
exactly compute the partition function of the four-fermi theory for some
small-$N$ cases. In Section 5, we compute the partition function of the four-
fermi theory for large-$N$ by applying an approximation using the Schwinger-
Dyson equation for two-point functions, and obtain analytic expressions of the
eigenvalue/vector distributions. In Section 6, we perform some Monte Carlo
simulations. The exact computations for small-$N$ are shown to precisely agree
with the Monte Carlo results. On the other hand, we find that the approximate
expressions for large-$N$ agree well with the Monte Carlo results, if the
expressions are multiplied by an extra factor depending on $N$. The final
section is devoted to a summary and future problems.

##  2 Real tensor eigenvalue/vector distributions

In this paper, we restrict ourselves to the symmetric real tensors with three
indices, $C_{abc}=C_{bac}=C_{bca}\in\mathbb{R}\ (a,b,c=1,2,\ldots,N)$, as the
simplest case. There are several different definitions of tensor
eigenvalues/vectors [11, 12, 13]. In this paper we employ the definition that
real eigenvectors of a given $C$ are defined by $v$’s satisfying

$\displaystyle C_{abc}v_{b}v_{c}=v_{a},\ v\in\mathbb{R}^{N}.$ (1)

Here repeated indices are assumed to be summed over, as will also be assumed
throughout this paper. Then the real eigenvector distribution for a given $C$
is given by

$\displaystyle\rho(v,C)=\sum_{i=1}^{N_{\rm
sol}(C)}\prod_{a=1}^{N}\delta(v_{a}-v_{a}^{i}),$ (2)

where $v^{i}\ (i=1,2,\ldots,N_{\rm sol}(C))$ are all the solutions of (1).

The expression (2) can be rewritten as

$\displaystyle\rho(v,C)=\left|\det
M\right|\,\prod_{a=1}^{N}\delta\left(v_{a}-C_{abc}v_{b}v_{c}\right),$ (3)

where $\det$ represents taking the determinant, $|\cdot|$ is to take the
absolute value, and $M$ is a matrix defined by

$\displaystyle M_{ab}=\frac{\partial}{\partial
v_{a}}\left(v_{b}-C_{bcd}v_{c}v_{d}\right)=\delta_{ab}-2C_{abc}v_{c}.$ (4)

Suppose the tensor $C$ is randomly distributed with the Gaussian distribution.
Then the distribution of $v$ is given by

$\displaystyle\rho(v)=\langle\rho(v,C)\rangle_{C}=A^{-1}\int_{\mathbb{R}^{D_{C}}}dC\,e^{-\alpha
C^{2}}\,\rho(v,C),$ (5)

where $dC=\prod_{a\leq b\leq c=1}^{N}dC_{abc}$, $C^{2}=C_{abc}C_{abc}$,
$\alpha$ is a positive number, $A=\int_{\mathbb{R}^{D_{C}}}dC\,e^{-\alpha
C^{2}}$, and $D_{C}=N(N+1)(N+2)/6$, i.e., the total number of independent
components of $C$.

In the previous paper [18], we computed a similar quantity which has $\det M$
instead of $|\det M|$ in (3). The previous quantity was easier to compute,
since it can simply be expressed by a couple of fermions by using the formula,
$\det M=\int d{\bar{\psi}}d\psi\,e^{{\bar{\psi}}_{a}M_{ab}\psi_{b}}$ [19]. On
the other hand, what makes the expression (3) difficult to deal with is that
$|\det M|$ is not analytic. One way to turn it to an analytic expression is to
consider a quantity,

$\displaystyle\rho(v,C,R,\epsilon)=\left(\det\left(M^{2}+\epsilon
I\right)\right)^{R}\prod_{a=1}^{N}\delta\left(v_{a}-C_{abc}v_{b}v_{c}\right),$
(6)

where $\epsilon>0$ is a regularization parameter, and $I$ is the identity
matrix, $I_{ab}=\delta_{ab}$. Then, for random $C$ with the Gaussian
distribution, we have

$\displaystyle\rho(v,R,\epsilon)=\langle\rho(v,C,R,\epsilon)\rangle_{C}=A^{-1}\int_{\mathbb{R}^{D_{C}}}dC\,e^{-\alpha
C^{2}}\left(\det\left(M^{2}+\epsilon
I\right)\right)^{R}\prod_{a=1}^{N}\delta\left(v_{a}-C_{abc}v_{b}v_{c}\right).$
(7)

The expression corresponding to (5) can be obtained by putting $R=1/2$ and
taking the $\epsilon\rightarrow+0$ limit:

$\displaystyle\rho(v)=\rho(v,1/2,+0).$ (8)

As we will find in Section 5, the regularization parameter $\epsilon$ is
necessary for the large-$N$ approximate computation, to obtain an expression
which is valid in all the region of $v$ and to take the
$\epsilon\rightarrow+0$ limit unambiguously: Directly starting from the
expression with $\epsilon=0$ has a singularity and it is not clear how the
expression should be extended to the full region.

The real eigenvalues accompanied with real eigenvectors (Z-eigenvalues in the
terminology of [11]) for a real symmetric order-three tensor $C$ are the $z$’s
satisfying

$\displaystyle C_{abc}w_{b}w_{c}=z\,w_{a},$ (9)

where $z\in\mathbb{R}$, $w\in\mathbb{R}^{N}$with $|w|=1$
($|w|=\sqrt{w_{a}w_{a}}$). The relation to (1) is $w=v/|v|$ and $z=1/|v|$.
Therefore, once we obtain $\rho(v)$, the Z-eigenvalue distribution is obtained
by

$\displaystyle\begin{split}\rho_{\rm
eigenvalue}(z)&=\rho(v)S^{N-1}|v|^{N-1}\frac{d|v|}{dz}\\\
&=\rho(1/z)S_{N-1}z^{-N-1},\end{split}$ (10)

where $S_{N-1}=2\pi^{N/2}/\Gamma(N/2)$ is the surface volume of a unit sphere
in an $N$-dimensional space. To derive this expression, we have used that
$\rho(v)$ actually depends only on $|v|$, because of the
$O(N)$-invariance444The system is invariant under the orthogonal
transformation in the vector space associated to the lower index. of (5), and
have used $d^{N}v=S_{N-1}|v|^{N-1}d|v|$. Note that we have abusively written
$1/z$ as the argument on the righthand side of (10) to represent an arbitrary
vector of size $1/z$.

##  3 A four-fermi theory with $R$ replicas

By using the well-known formulas, $\int_{\mathbb{R}}dx\,e^{ixy}=2\pi\delta(y)$
and $\int d{\bar{\psi}}d\psi\,e^{{\bar{\psi}}_{a}A_{ab}\psi_{b}}=\det A$ [19],
the expression (7) can be rewritten as

$\displaystyle\rho(v,R,\epsilon)=A^{-1}(2\pi)^{-N}\int dCd\lambda
d\bar{\psi}d\psi\,e^{S_{1}},$ (11)

where

$\displaystyle S_{1}=-\alpha
C^{2}+i\lambda_{a}(v_{a}-C_{abc}v_{b}v_{c})+{\bar{\psi}}_{a}^{i}M^{2}_{ab}\psi_{bi}+\epsilon{\bar{\psi}}_{a}^{i}\psi_{ai}.$
(12)

Here we have assumed $R$ to be integer, $\bar{\psi}_{a}^{i},\psi_{ai}\
(a=1,2,\ldots,N;\ i=1,2,\ldots,R)$ are fermionic, and $\lambda_{a}\
(a=1,2,\ldots,N)$ are bosonic with the integration region of $\mathbb{R}^{N}$.
Note that the paired new indices (namely, $i$) are also assumed to be summed
over in the above expression, as will be assumed throughout this paper. Note
also that the system (11) is invariant under the following $GL(R)$
transformation concerning the new index:

$\displaystyle\begin{split}\psi^{\prime}{}_{ai}&=G{}_{i}{}^{i^{\prime}}\psi_{ai^{\prime}},\
\bar{\psi}^{\prime}{}_{a}^{i}=\bar{\psi}^{i^{\prime}}_{a}G^{-1}{}_{i^{\prime}}{}^{i},\end{split}$
(13)

for where $G\in GL(R)$.

In the similar manner as the procedure taken in the previous paper [18], we
want to first integrate over the bosonic variables, $C$ and $\lambda$, to
obtain a fermionic theory. However, $M^{2}$ contains $C$ in a quadratic manner
and is not straightforward to deal with. To circumvent this matter, let us
introduce another pair of fermions $\bar{\varphi},\varphi$ to rewrite (12)
into an expression linear in $M$:

$\displaystyle\rho(v,R,\epsilon)=A^{-1}(2\pi)^{-N}(-1)^{NR}\int dCd\lambda
d\bar{\psi}d\psi d\bar{\varphi}d\varphi\,e^{S_{2}},$ (14)

where

$\displaystyle\begin{split}S_{2}=-\alpha
C^{2}+&i\lambda_{a}(v_{a}-C_{abc}v_{b}v_{c})-\bar{\varphi}_{a}^{i}\varphi_{ai}+\epsilon\bar{\psi}_{a}^{i}\psi_{ai}-\bar{\psi}_{a}^{i}M_{ab}\varphi_{bi}-\bar{\varphi}_{a}^{i}M_{ab}\psi_{bi}.\end{split}$
(15)

The equivalence between (11) and (14) can be shown by

$\displaystyle\begin{split}\int
d\bar{\varphi}d\varphi\,e^{-\bar{\varphi}_{a}\varphi_{a}-\bar{\psi}_{a}M_{ab}\varphi_{b}-\bar{\varphi}_{a}M_{ab}\psi_{b}}&=\int
d\bar{\varphi}d\varphi\,e^{-(\bar{\varphi}_{a}+\bar{\psi}_{b}M_{ba})(\varphi_{a}+M_{ac}\psi_{c})+\bar{\psi}_{a}M^{2}_{ab}\psi_{b}}\\\
&=(-1)^{N}e^{\bar{\psi}_{a}M^{2}_{ab}\psi_{b}}.\end{split}$ (16)

Now let us first integrate over $C$, assuming that the final result does not
depend on this change of the order of the integrations. The terms containing
$C$ in (15) are

$\displaystyle S_{C}=-\alpha
C^{2}-iC_{abc}\lambda_{a}v_{b}v_{c}+2C_{abc}\bar{\psi}_{a}^{i}\varphi_{bi}v_{c}+2C_{abc}\bar{\varphi}_{a}^{i}\psi_{bi}v_{c}.$
(17)

Therefore the integration over $C$ is just a Gaussian integration and we
obtain

$\displaystyle\int_{\mathbb{R}^{D_{C}}}dCe^{S_{C}}=A\,e^{\delta S_{C}},$ (18)

where

$\displaystyle\begin{split}\delta
S_{C}&=\frac{1}{\alpha}\left(\frac{1}{6}\sum_{\sigma}\left(-\frac{i}{2}\lambda_{\sigma_{a}}v_{\sigma_{b}}v_{\sigma_{c}}+\bar{\psi}^{i}_{\sigma_{a}}\varphi_{\sigma_{b}i}v_{\sigma_{c}}+\bar{\varphi}^{i}_{\sigma_{a}}\psi_{\sigma_{b}i}v_{\sigma_{c}}\right)\right)^{2}.\end{split}$
(19)

Here the summation over $\sigma$ is over all the permutations of $a,b,c$, the
necessity of which comes from that $C$ is a symmetric tensor. Explicitly
expanding the expression (19), we obtain

$\displaystyle\delta
S_{C}=-\frac{|v|^{4}}{12\alpha}\lambda_{a}B_{ab}\lambda_{b}-i\lambda_{a}D_{a}+E$
(20)

with

$\displaystyle B_{ab}$
$\displaystyle=\delta_{ab}+2\hat{v}_{a}\hat{v}_{b}=I_{\perp\,ab}+3I_{\parallel\,ab},$
(21) $\displaystyle D_{a}$
$\displaystyle=\frac{|v|^{3}}{3\alpha}\left({\bar{\psi}}_{\parallel}^{i}{\varphi}_{\parallel}{}_{i}+\bar{\varphi}_{\parallel}^{i}\psi_{\parallel}{}_{i}\right)\hat{v}_{a}+\frac{|v|^{3}}{3\alpha}\left({\bar{\psi}}_{a}^{i}{\varphi}_{\parallel}{}_{i}+{\bar{\psi}}_{\parallel}^{i}{\varphi}_{ai}+\bar{\varphi}_{a}^{i}\psi_{\parallel}{}_{i}+\bar{\varphi}_{\parallel}^{i}\psi_{ai}\right),$
(22) $\displaystyle E$
$\displaystyle=\frac{1}{\alpha}\left(\frac{1}{6}\sum_{\sigma}\left({\bar{\psi}}_{\sigma_{a}}^{i}{\varphi}_{\sigma_{b}i}v_{\sigma_{c}}+\bar{\varphi}_{\sigma_{a}}^{i}\psi_{\sigma_{b}i}v_{\sigma_{c}}\right)\right)^{2},$
(23)

where $\hat{v}_{a}=v_{a}/|v|$, and $\parallel$ denotes the projection to
$\hat{v}$, $\psi_{\parallel}=\psi_{a}\hat{v}_{a}$, and so on. As above, the
matrix $B$ can be rewritten as a linear sum of the projection operators, where
$I_{\parallel}$ is the projection operator to the one-dimensional linear
subspace spanned by $\hat{v}$, and $I_{\perp}$ is to the subspace transverse
to $\hat{v}$.

From (15) and (20), the remaining terms containing $\lambda$ are

$\displaystyle
S_{\lambda}=-\frac{|v|^{4}}{12\alpha}\lambda_{a}B_{ab}\lambda_{b}+i\lambda_{a}(v_{a}-D_{a}).$
(24)

Integrating over $\lambda$ gives

$\displaystyle\int_{\mathbb{R}^{N}}d\lambda\
e^{S_{\lambda}}=(12\pi\alpha)^{\frac{N}{2}}|v|^{-2N}(\det
B)^{-\frac{1}{2}}e^{\delta S_{\lambda}}$ (25)

with

$\displaystyle\begin{split}\delta
S_{\lambda}&=-3\alpha|v|^{-4}(v_{a}-D_{a})B^{-1}_{ab}(v_{b}-D_{b})\\\
&=-3\alpha|v|^{-4}\left(D_{\perp}\cdot
D_{\perp}+\frac{1}{3}(|v|-D_{\parallel})^{2}\right)\\\
&=-\frac{\alpha}{|v|^{2}}+2\left({\bar{\psi}}_{\parallel}^{i}{\varphi}_{\parallel}^{i}+\bar{\varphi}_{\parallel}^{i}\psi_{\parallel}^{i}\right)-3\alpha|v|^{-4}\left(D_{\perp}\cdot
D_{\perp}+\frac{1}{3}D_{\parallel}^{2}\right),\end{split}$ (26)

where we have used $B^{-1}=I_{\perp}+\frac{1}{3}I_{\parallel}$, and from (22)
the projections of $D$ to the parallel and transverse directions to $\hat{v}$
are respectively given by

$\displaystyle\begin{split}D_{\perp}&=\frac{|v|^{3}}{3\alpha}\left(\bar{\psi}_{\perp}^{i}{\varphi}_{\parallel}{}_{i}+{\bar{\psi}}_{\parallel}^{i}\varphi_{\perp}{}_{i}+\bar{\varphi}_{\perp}^{i}\psi_{\parallel}{}_{i}+\bar{\varphi}_{\parallel}^{i}\psi_{\perp}{}_{i}\right),\\\
D_{\parallel}&=\frac{|v|^{3}}{\alpha}\left({\bar{\psi}}_{\parallel}^{i}{\varphi}_{\parallel}{}_{i}+\bar{\varphi}_{\parallel}^{i}\psi_{\parallel}{}_{i}\right).\end{split}$
(27)

While the last term of (26) gives some four-fermi interaction terms, the
second term gives some corrections to the quadratic terms in the parallel
direction.

Collecting all the results above and using $\det B=3$, we obtain a four-fermi
theory,

$\displaystyle\rho(v,R,\epsilon)=3^{\frac{N-1}{2}}\pi^{-\frac{N}{2}}\alpha^{\frac{N}{2}}|v|^{-2N}e^{-\frac{\alpha}{|v|^{2}}}(-1)^{NR}\int
d{\bar{\psi}}d\psi d\bar{\varphi}d{\varphi}\,e^{K+K_{\parallel}+V},$ (28)

where the quadratic term $K,\,K_{\parallel}$ and the four-fermi interaction
terms $V$ are respectively given by

$\displaystyle\begin{split}K&=\left(\begin{array}[]{cc}\bar{\psi}_{\perp}^{i}&\bar{\varphi}_{\perp}^{i}\end{array}\right)\left(\begin{array}[]{cc}\epsilon&-1\cr-1&-1\end{array}\right)\left(\begin{array}[]{c}\psi_{\perp}{}_{i}\cr\varphi_{\perp}{}_{i}\end{array}\right),\\\
K_{\parallel}&=\left(\begin{array}[]{cc}{\bar{\psi}}_{\parallel}^{i}&\bar{\varphi}_{\parallel}^{i}\end{array}\right)\left(\begin{array}[]{cc}\epsilon&1\cr
1&-1\end{array}\right)\left(\begin{array}[]{c}\psi_{\parallel}{}_{i}\cr{\varphi}_{\parallel}{}_{i}\end{array}\right),\\\
V&=-3\alpha|v|^{-4}\left(D_{\perp}\cdot
D_{\perp}+\frac{1}{3}D_{\parallel}^{2}\right)+E\\\
&=-\frac{|v|^{2}}{6\alpha}\Big{(}\bar{\psi}_{\perp}^{i}\cdot\bar{\psi}_{\perp}^{j}\varphi_{\perp}{}_{i}\cdot\varphi_{\perp}{}_{j}+\bar{\psi}_{\perp}^{i}\cdot\varphi_{\perp}{}_{j}\bar{\psi}_{\perp}^{j}\cdot\varphi_{\perp}{}_{i}+\bar{\varphi}_{\perp}^{i}\cdot\bar{\varphi}_{\perp}^{j}\psi_{\perp}{}_{i}\cdot\psi_{\perp}{}_{j}\\\
&\ \ \ \ \ \ \ \ \ \ \ \
+\bar{\varphi}_{\perp}^{i}\cdot\psi_{\perp}{}_{j}\bar{\varphi}_{\perp}^{j}\cdot\psi_{\perp}{}_{i}+2\bar{\psi}_{\perp}^{i}\cdot\bar{\varphi}_{\perp}^{j}\varphi_{\perp}{}_{i}\cdot\psi_{\perp}{}_{j}+2\bar{\psi}_{\perp}^{i}\cdot\psi_{\perp}{}_{j}\bar{\varphi}_{\perp}^{j}\cdot\varphi_{\perp}{}_{i}\Big{)}.\end{split}$
(29)

What is surprising in $V$ in (29) is that the parallel components are not
contained. Therefore, the parallel components are free theory and can
trivially be integrated over:

$\displaystyle\int
d{\bar{\psi}}_{\parallel}d\psi_{\parallel}d\bar{\varphi}_{\parallel}d{\varphi}_{\parallel}\,e^{K_{\parallel}}=(-1)^{R},$
(30)

where we have already taken the $\epsilon\rightarrow+0$ limit, since this is
smooth. Then we finally obtain a four-fermi theory containing only the
transverse components,

$\displaystyle\rho(v,R,\epsilon)=3^{\frac{N-1}{2}}\pi^{-\frac{N}{2}}\alpha^{\frac{N}{2}}|v|^{-2N}e^{-\frac{\alpha}{|v|^{2}}}(-1)^{(N-1)R}\int
d\bar{\psi}_{\perp}d\psi_{\perp}d\bar{\varphi}_{\perp}d\varphi_{\perp}\,e^{K+V}.$
(31)

##  4 Exact computations for small-$N$

Simple crosschecks of the formula, (31) with (29), can be performed by exactly
computing it. In the $N=1$ case, we have no transverse directions, and hence
(31) gives

$\displaystyle\rho_{N=1}(v,R,0)=\pi^{-\frac{1}{2}}\alpha^{\frac{1}{2}}v^{-2}e^{-\frac{\alpha}{|v|^{2}}}.$
(32)

Indeed, from that the eigenvector is given by $v=1/C$ for $N=1$, the Gaussian
distribution of $C$ is the same as (32), because
$\pi^{-\frac{1}{2}}\alpha^{\frac{1}{2}}e^{-\alpha
C^{2}}dC=\rho_{N=1}(v,R,0)dv$.

Another obvious consequence is that, for integer $R$, the four-fermi theory
(31) can in principle be computed by expanding the exponential in $K,V$:

$\displaystyle\int
d\bar{\psi}_{\perp}d\psi_{\perp}d\bar{\varphi}_{\perp}d\varphi_{\perp}\,e^{K+V}=\sum_{n=0}^{(N-1)R}\frac{1}{n!(2(N-1)R-2n)!}\int
d\bar{\psi}_{\perp}d\psi_{\perp}d\bar{\varphi}_{\perp}d\varphi_{\perp}\,V^{n}\,K^{2(N-1)R-2n}.$
(33)

This formula comes from the fact that each term of the integrand must be a
product of exactly $4(N-1)R$ fermions for the fermionic integral to be non-
zero [19]. The $\epsilon\rightarrow+0$ limit of (33) is straightforward, and,
for integer $R$, (31) generally has the form,

$\displaystyle\rho(v,R,0)=3^{\frac{N-1}{2}}\pi^{-\frac{N}{2}}\alpha^{\frac{N}{2}}|v|^{-2N}e^{-\frac{\alpha}{|v|^{2}}}{\cal
L}_{N,R},$ (34)

where ${\cal L}_{N,R}$ is a polynomial function of $x=|v|^{2}/(6\alpha)$,
${\cal L}_{N,R}=1+c_{1}x+c_{2}x^{2}+\cdots+c_{(N-1)R}x^{(N-1)R}$, where
$c_{i}$ are some coefficients depending on $N,R$. Then, as in (8), $\rho(v)$
can in principle be obtained by taking the analytic continuation of ${\cal
L}_{N,R}$ to $R=1/2$.

However, the exact computation of ${\cal L}_{N,R}$ for general $N,R$ seems to
be a difficult task. It becomes more and more cumbersome very quickly, as
$N,R$ increase. Still it is doable for some small $N,R$ by using computers. By
a Mathematica package for Grassmann variables [20], we obtain

$\displaystyle\begin{split}{\cal L}_{N=2,R=1}&=1+4x,\\\ {\cal
L}_{N=3,R=1}&=1+4x+28x^{2},\\\ {\cal L}_{N=2,R=2}&=1+24x+48x^{2},\\\ {\cal
L}_{N=3,R=2}&=1+40x+552x^{2}+1248x^{3}+5136x^{4}.\end{split}$ (35)

The formula (34) with (35) will be checked by Monte Carlo simulations in
Section 6.

##  5 Approximate computations for large-$N$

As was demonstrated in the last part of Section 4, the expression (31) can in
principle be computed for general integer $N,R$, and the $R=1/2$ limit can be
taken by analytic continuation. However, this does not seem straightforward in
practice. Therefore, in this section, we apply the Schwinger-Dyson equation to
the four-fermi theory, assuming the factorization of the four-point
correlation functions into the products of two-point functions. A self-
contained brief explanation of the approximation is given in Appendix A.

The four-fermi theory (31) has the following symmetry, $O(N-1)\times GL(R)$,
for the transverse variables $X,\,\bar{X}\ (X=\psi_{\perp},\varphi_{\perp})$:

$\displaystyle\begin{split}X^{\prime}{}_{ai}&=g_{1}{}_{a}{}^{a^{\prime}}g_{2}{}_{i}{}^{i^{\prime}}X_{a^{\prime}i^{\prime}},\
\bar{X}^{\prime}{}_{a}^{i}=g_{1}{}_{a}{}^{a^{\prime}}\bar{X}^{i^{\prime}}_{a^{\prime}}g_{2}^{-1}{}_{i^{\prime}}{}^{i},\end{split}$
(36)

where $g_{1}$ belongs to the defining representation of $O(N-1)$ in the
subspace transverse to $\hat{v}$, and $g_{2}$ belongs to the defining
representation of $GL(R)$. Assuming that these symmetries do not break down in
the large-$N$ limit, we can assume the following forms of the two point
functions:

$\displaystyle\begin{split}\langle\bar{\psi}_{\perp}{}_{a}^{i}\psi_{\perp}{}_{bj}\rangle&=Q_{11}I_{\perp\,ab}\,\delta^{i}_{j},\\\
\langle\bar{\psi}_{\perp}{}_{a}^{i}\varphi_{\perp}{}_{bj}\rangle&=Q_{12}I_{\perp\,ab}\,\delta^{i}_{j},\\\
\langle\bar{\varphi}_{\perp}{}_{a}^{i}\psi_{\perp}{}_{bj}\rangle&=Q_{21}I_{\perp\,ab}\,\delta^{i}_{j},\\\
\langle\bar{\varphi}_{\perp}{}_{a}^{i}\varphi_{\perp}{}_{bj}\rangle&=Q_{22}I_{\perp\,ab}\,\delta^{i}_{j},\\\
\hbox{Others}&=0,\end{split}$ (37)

where $Q_{ij}$ are some numbers. From (69) in Appendix A, the effective action
written in terms of $Q_{ij}$ is given by

$\displaystyle\begin{split}S_{\rm eff}&=\langle K+V\rangle-(N-1)R\log(-\det
Q)\\\ &=(N-1)R\left(\epsilon
Q_{11}-Q_{12}-Q_{21}-Q_{22}-\frac{|v|^{2}(N-1)}{6\alpha}\left(Q_{12}^{2}+Q_{21}^{2}+2Q_{11}Q_{22}\right)-\log(-\det
Q)\right)\end{split}$ (38)

in the leading order of $N-1$. Here the minus sign in the logarithm is
appropriate for the current computation, since the solution of $Q_{ij}$ below
gives $\det Q<0$. In the derivation of (38), we have used the factorizations
into two-point functions and (37), such as

$\displaystyle\begin{split}\langle\bar{\psi}_{\perp}^{i}\cdot\varphi_{\perp}{}_{j}\bar{\psi}_{\perp}^{j}\cdot\varphi_{\perp}{}_{i}\rangle&=\langle\bar{\psi}_{\perp}^{i}{}_{a}\varphi_{\perp}{}_{aj}\rangle{}\langle\bar{\psi}_{\perp}^{j}{}_{b}\varphi_{\perp}{}_{bi}\rangle-\langle\bar{\psi}_{\perp}^{i}{}_{a}\varphi_{\perp}{}_{bi}\rangle{}\langle\bar{\psi}_{\perp}^{j}{}_{b}\varphi_{\perp}{}_{aj}\rangle\\\
&=(N-1)^{2}RQ_{12}^{2}-(N-1)R^{2}Q_{12}^{2},\end{split}$ (39)

and have taken only the leading order terms in $N-1$.

The stationary equations,

$\displaystyle\frac{\partial S_{\rm eff}}{\partial Q_{11}}=\frac{\partial
S_{\rm eff}}{\partial Q_{12}}=\frac{\partial S_{\rm eff}}{\partial
Q_{21}}=\frac{\partial S_{\rm eff}}{\partial Q_{22}}=0,$ (40)

have four independent solutions, and, among them, we should choose a solution
which reproduces the $v=0$ limit (this is a free theory limit, since $V=0$)
given by

$\displaystyle Q_{11}=\frac{1}{1+\epsilon},\
Q_{12}=Q_{21}=\frac{-1}{1+\epsilon},\
Q_{22}=\frac{\epsilon}{1+\epsilon},\hbox{ for }v=0.$ (41)

The explicit expression of the solution is given in Appendix B with
$x=v^{2}(N-1)/(3\alpha)$. The $\epsilon\rightarrow+0$ limit has a singularity
at $x=1/4$, and for each region, we obtain for $\epsilon\rightarrow+0$

  * •

$0<x<1/4$

$\displaystyle\begin{split}&Q_{11}=\frac{-\sqrt{1-4x}+1}{2x\sqrt{1-4x}},\\\
&Q_{12}=Q_{21}=\frac{1-\sqrt{1-4x}-4x}{2x\sqrt{1-4x}},\\\
&Q_{22}=0,\end{split}$ (42)

  * •

$x>1/4$

$\displaystyle\begin{split}&Q_{11}=\frac{\sqrt{-1+4x}}{2\sqrt{\epsilon}x}-\frac{1}{2x}+{\cal
O}(\sqrt{\epsilon}),\\\
&Q_{12}=Q_{21}=-\frac{1}{2x}+\frac{\sqrt{\epsilon}}{2x\sqrt{-1+4x}}+{\cal
O}(\epsilon^{\frac{3}{2}}),\\\
&Q_{22}=-\frac{\sqrt{\epsilon}\sqrt{-1+4x}}{2x}+\frac{\epsilon}{2x}+{\cal
O}(\epsilon^{\frac{3}{2}}).\end{split}$ (43)

The latter solution is not well-defined in $\epsilon\rightarrow+0$, but the
$S_{\rm eff}$ for those solutions has well-defined limits in both regions:

$\displaystyle S^{\epsilon=+0}_{\rm
eff}(x)=\left\\{\begin{array}[]{ll}\frac{(N-1)R}{2}\left(2+\log(16)+\frac{1-\sqrt{1-4x}}{x}-4\log\left(1-\sqrt{1-4x}\right)+4\log(x)\right)&\hbox{
for }0<x<\frac{1}{4},\\\ \frac{(N-1)R}{2x}(1+2x+2x\log(x))&\hbox{ for
}\frac{1}{4}<x.\end{array}\right.$ (46)

Then from (71) we obtain for large-$N$

$\displaystyle\rho(v,R,+0)=3^{\frac{N-1}{2}}\pi^{-\frac{N}{2}}\alpha^{\frac{N}{2}}|v|^{-2N}e^{-\frac{\alpha}{|v|^{2}}}e^{S^{\epsilon=+0}_{\rm
eff}(x)-2(N-1)R},$ (47)

where the subtraction in the exponent is due to $S^{\epsilon=+0}_{\rm
eff}(0)=2(N-1)R$.

Finally, we want to argue that only the expression at $x>1/4$ in (46)
practically matters, when $N$ is large. The reason is as follows. Because of
the exponential damping factor $e^{-\frac{\alpha}{|v|^{2}}}$, $\rho(v,R,+0)$
takes significant values only in the region $|v|\gtrsim\sqrt{\alpha}$. This
means that the relevant region satisfies
$x=|v|^{2}(N-1)/(3\alpha)\gtrsim(N-1)/3>1/4$ for large $N$. Therefore, we can
practically use $S_{\rm eff}^{\epsilon=+0}$ at $x>1/4$ for all the region of
$x$ for large-$N$. By doing this, we finally obtain, for large-$N$ and
$R=1/2$,

$\displaystyle\rho(v)=\rho(v,1/2,+0)=(N-1)^{\frac{N-1}{2}}e^{-\frac{N-1}{2}}\pi^{-\frac{N}{2}}\alpha^{\frac{1}{2}}|v|^{-N-1}e^{-\frac{\alpha}{4|v|^{2}}}.$
(48)

The result (48) is for the distribution of vector $v$. For comparison with
Monte Carlo simulations, it is more convenient to transform it to the
distribution of the size $|v|$. Since $d^{N}v=S_{N-1}|v|^{N-1}d|v|$ with
$S_{N-1}=2\pi^{N/2}/\Gamma(N/2)$ being the surface volume of a unit sphere in
$N$-dimensions, we obtain for large-$N$

$\displaystyle\rho_{\rm
size}(|v|)=\frac{2(N-1)^{\frac{N-1}{2}}e^{-\frac{N-1}{2}}\alpha^{\frac{1}{2}}}{\Gamma\left(\frac{N}{2}\right)}|v|^{-2}e^{-\frac{\alpha}{4|v|^{2}}}.$
(49)

One can also transform the result to the real eigenvalue distribution
(Z-eigenvalues of [11]), using (10): For large-$N$,

$\displaystyle\rho_{\rm
eigevalue}(z)=\frac{2(N-1)^{\frac{N-1}{2}}e^{-\frac{N-1}{2}}\alpha^{\frac{1}{2}}}{\Gamma\left(\frac{N}{2}\right)}e^{-\frac{\alpha}{4}z^{2}}.$
(50)

Interestingly, this real eigenvalue distribution is just given by the Gaussian
function in this approximation.

##  6 Comparison with Monte Carlo simulations

In this section, we perform some Monte Carlo simulations, and compare the
results with the exact small-$N$ and the approximate large-$N$ results, which
are respectively obtained in Sections 4 and 5.

The eigenvector equations (1) are a system of polynomial equations, which can
be solved by appropriate polynomial equation solvers. We use Mathematica 12
for this purpose. It generally produces all the solutions including complex
ones, and we pick up all the real ones out of them. Whether all the real
eigenvectors are covered can be guaranteed by checking whether the number of
all the generally complex solutions agrees with the known formula, $2^{N}-1$
[13].

With the above way of obtaining real eigenvectors, our procedure of numerical
simulations can be summarized as follows.

  * •

Randomly generate $C$ with the Gaussian distribution: Each independent
component is randomly generated by $C_{ijk}=\sigma_{ijk}/\sqrt{d(i,j,k)},\
(i\leq j\leq k)$. Here $\sigma_{ijk}$ is a random number following the normal
distribution of mean value zero and standard deviation one. $d(i,j,k)$ is a
degeneracy factor defined by

$\displaystyle\begin{split}d(i,j,k)=\left\\{\begin{array}[]{ll}1,&i=j=k\\\
3,&i=j\neq k,\,i\neq j=k,\,k=i\neq j\\\ 6,&i\neq k\neq j\neq
i\end{array}\right..\end{split}$ (51)

The above distribution corresponds to choosing $\alpha=1/2$ in (5), since

$\displaystyle\begin{split}C^{2}=C_{abc}C_{abc}=\sum_{i\leq j\leq
k=1}^{N}d(i,j,k)\,C_{ijk}^{2}\end{split}$ (52)

due to $C$ being a symmetric tensor.

  * •

Pick up all the real eigenvectors by the way mentioned above.

  * •

Store the size $|v|$ and the value of $\det M$ of each of all the real
eigenvectors.

  * •

Repeat the above processes.

By the above repeating procedure we obtain a data of $(|v_{i}|,\det M_{i})\
(i=1,2,\ldots,L)$, where $L$ is the total number of real eigenvectors
generated. For the general case of $R$, we define

$\displaystyle\rho_{R}^{\rm sim}((k+1/2)\delta v)=\frac{1}{\delta
vN_{C}}\sum_{i=1}^{L}|\det M|^{2R-1}\,\theta(k\delta v<|v_{i}|\leq(k+1)\delta
v),$ (53)

where $N_{C}$ denotes the total number of randomly generated $C$, $\delta v$
is a bin size, $k=0,1,2,\ldots$, and $\theta$ is a support function which
takes 1 if the inequality of the argument is satisfied, but zero otherwise.
This quantity corresponds to $\rho(v,R,0)S_{N-1}|v|^{N-1}$ with $\alpha=1/2$
of the analytical result by a derivation similar to that of (49). The $-1$ in
the exponent of $|\det M|$ in (53) comes from the consideration of the measure
associated to the delta functions, namely, the difference between (2) and (3).

Figure 1:  Some non-trivial checks of the general formula (31) with (29). The
analytical results, (34) with (35) for $\alpha=1/2$, (solid lines) and the
results from the Monte Carlo simulations (53) (dots) for $(N,R)=(2,1)$,
$(3,1)$, $(2,2)$, $(3,2)$ from the left to the right panels are compared.
$\delta v=0.03$. $N_{C}=3\cdot 10^{4}$ for the former two, and $N_{C}=10^{5}$
for the latter two.

In Figure 1, the numerical datas (53) for $N=2,3,\ R=2,3$, and the analytical
results, $\rho(v,R,0)S_{N-1}|v|^{N-1}$ with $\rho(v,R,0)$ being given by (34)
with (35) for $\alpha=1/2$, are compared. They precisely agree. The agreement
includes the allover numerical factor, and gives non-trivial checks of the
general formula (31) with (29).

For $R=1/2$ in (53), we have

$\displaystyle\rho_{\rm size}^{\rm sim}((k+1/2)\delta v)=\frac{1}{\delta
vN_{C}}\sum_{i=1}^{L}\theta(k\delta v<|v_{i}|\leq(k+1)\delta v).$ (54)

For large-$N$, this is the numerical quantity corresponding to (49) with
$\alpha=1/2$.

Figure 2:  Left: (54) from a Monte Carlo simulation for $N=14$ is plotted by
dots. $N_{C}=400,\ \delta v=0.01$. This is compared with $C\rho_{\rm size}$
from (49) with $C=5.27$ (the best fitting value), which is plotted by a solid
line. Right: The overall factors $C$ needed for the best fitting for various
$N$ are plotted by dots. The fitting line is $-3.6+3.2N^{0.38}$.

In the left panel of Figure 2 the analytical result (49) and the numerical
result (54) are compared. In this figure, the analytical result is multiplied
by an extra allover factor, which means that the overall factor is not
correctly computed in the analytical result, while the functional form agrees
well with the numerical data. As shown in the right panel, the extra factor
$C$ needed to have good agreement is dependent on $N$. As far as the fitting
line implies, the factor seems to asymptotically diverge in
$N\rightarrow\infty$.

Figure 3:  The numerical result (55) and the analytical result (50) for
large-$N$ are compared for $N=14$. $N_{C}=400,\ \delta v=0.02$, and $C=5.6$
(the best fitting value).

In a similar manner, one can obtain the real eigenvalue (Z-eigenvalue)
distribution from the numerical data by using the relation $z=1/|v|$ mentioned
in the lines above (10):

$\displaystyle\rho_{\rm eigenvalue}^{\rm sim}((k+1/2)\delta z)=\frac{1}{\delta
zN_{C}}\sum_{i=1}^{L}\theta\left(k\delta z<\frac{1}{|v_{i}|}\leq(k+1)\delta
z\right).$ (55)

For large-$N$ this corresponds to the analytic expression (50), and they are
compared in Figure 3. We again find good agreement on the functional form,
which is simply the Gaussian.

The origin of the mismatch of the overall factor may be corrected by the next-
leading order. As shown in (43), $Q_{ij}$ are in the order of $1/\sqrt{N}$.
Then the next-leading order corrections would be $\delta Q_{ij}\sim 1/N$ or
$1/N^{3/2}$. Putting this correction to (38) can produce a substantial
correction to $S^{\epsilon=+0}_{\rm eff}$, which can change the overall
factor. There are also some other potential sources of corrections, and it
will be a complex task to compute all of them, because of the many interaction
terms of $V$ in (29).

##  7 Summary and future problems

Considering the important roles of the eigenvalue distributions in matrix
models, it would be interesting to compute corresponding quantities in tensor
models and study their properties and roles. In this paper, we have studied
the real tensor eigenvalue/vector distributions for random real symmetric
order-three tensors with the Gaussian distribution. We first rewrote the
problem as the computation of a partition function of a four-fermi theory. We
exactly computed the partition function for some small-$N$ cases, and found
precise agreement with the Monte Carlo results. For large-$N$ we approximately
computed the partition function by using a Schwinger-Dyson equation for the
two-point functions. Interestingly the real eigenvalue distribution turned out
to be simply the Gaussian within this approximation. The comparison with the
Monte Carlo simulations showed that, by multiplying an overall numerical
factor depending on $N$, the approximate analytic expression is in good
agreement with the numerical results.

An obvious future problem is to improve the approximate computation of the
partition function of the four-fermi theory for large-$N$ so that the overall
factor is also correctly derived. In this paper, we used a Schwinger-Dyson
equation for the two-point functions, but it is feasible to extend it to
include higher-point functions, though this will be much more complicated
because of the presence of many terms of the four-fermi interactions. Another
possible direction is that, while we considered a replicated fermion system to
take analytic continuation to the wanted case, it is also possible to consider
a fermion-boson system to directly compute the distribution without analytic
continuation. It will be useful for future extended study to figure out which
ways of computations give correct results efficiently. It also seems important
to check whether the simple Gaussian distribution of the real eigenvalues is
robust against the improvement.

Another interesting direction will be to extend the Gaussian tensor model we
considered to more general tensor models. In particular, an interesting
phenomenon which occurs in matrix models is the topological change of
eigenvalue distributions. As for tensor models, first of all, it is unknown
whether there exists such a topological transition in tensor eigenvalue/vector
distributions, and, if it exists, how this is related to the dynamics of
tensor models is also an interesting question. It would be worth mentioning
that a similar phenomenon with topological changes of distributions of
dynamical variables was observed in the analysis of the wave function of a
tensor model in the Hamiltonian formalism [21, 22]. We would therefore expect
that such phenomena could be universal across matrix and tensor models, and
this could be studied by extending the current work to more general cases.

## Acknowledgements

The author is supported in part by JSPS KAKENHI Grant No.19K03825.

##  Appendix Appendix A Approximate computations with the Schwinger-Dyson
equation

In this appendix, we explain the approximation by using the Schwinger-Dyson
equation, which we employ in the text. The content of this section is rather
standard, and is therefore supposed to be for some unfamiliar readers.

The problem we are dealing with is a four-fermi theory which can generally be
written in the form,

$\displaystyle
S=\bar{\psi}_{a}K_{ab}\psi_{b}+gB_{abcd}\bar{\psi}_{a}\bar{\psi}_{b}\psi_{c}\psi_{d},$
(56)

where $g$ is a coupling, and $B$ is assumed to satisfy the partially
antisymmetry condition given by

$\displaystyle B_{abcd}=-B_{bacd}=-B_{abdc},$ (57)

due to the fermionic property of the variables. The action $S$ is invariant
under a scaling transformation,

$\displaystyle\bar{\psi}_{a}^{\prime}=s\bar{\psi}_{a},\
\psi_{a}^{\prime}=s^{-1}\psi_{a},$ (58)

where $s$ can be an arbitrary number. The partition function is defined by

$\displaystyle Z=\int d\bar{\psi}d\psi\,e^{S},$ (59)

where the integration measure is defined so that [19]

$\displaystyle\int d\bar{\psi}d\psi\,e^{\bar{\psi}_{a}K_{ab}\psi_{b}}=\det K.$
(60)

Since an integral of a total derivative identically vanishes for the fermionic
integration, we can write down the following consistency equation,

$\displaystyle\begin{split}0&=\int
d\bar{\psi}d\psi\frac{\partial}{\partial\bar{\psi}_{a}}(\bar{\psi}_{b}\,e^{S})\\\
&=\int
d\bar{\psi}d\psi\left(\delta_{ab}-K_{ac}\bar{\psi}_{b}\psi_{c}-2gB_{acde}\bar{\psi}_{b}\bar{\psi}_{c}\psi_{d}\psi_{e}\right)e^{S},\end{split}$
(61)

where the minus signs are due to the fermionic property. By dividing by $Z$,
we obtain a Schwinger-Dyson equation,

$\displaystyle\begin{split}K_{ac}\langle\bar{\psi}_{b}\phi_{c}\rangle+2gB_{acde}\langle\bar{\psi}_{b}\bar{\psi}_{c}\psi_{d}\psi_{e}\rangle=\delta_{ab},\end{split}$
(62)

where $\langle{\cal O}\rangle$ denotes correlation functions defined by

$\displaystyle\langle{\cal O}\rangle=\frac{1}{Z}\int d\bar{\psi}d\psi\,{\cal
O}e^{S}.$ (63)

The four-point function in (62) may be expanded in terms of connected
correlation functions,

$\displaystyle\langle\bar{\psi}_{b}\bar{\psi}_{c}\psi_{d}\psi_{e}\rangle=-\langle\bar{\psi}_{b}\psi_{d}\rangle\langle\bar{\psi}_{c}\psi_{e}\rangle+\langle\bar{\psi}_{b}\psi_{e}\rangle\langle\bar{\psi}_{c}\psi_{d}\rangle+\langle\bar{\psi}_{b}\bar{\psi}_{c}\psi_{d}\psi_{e}\rangle_{c},$
(64)

where $\langle{\cal O}\rangle_{c}$ denotes the connected correlation
functions. Here we have assumed that only the correlation functions with the
same number of $\bar{\psi}$ and $\psi$ remain non-zero due to the symmetry
(58). Now assuming that we can ignore the four-point connected correlation
functions as higher-order terms,
$\langle\bar{\psi}_{b}\bar{\psi}_{c}\psi_{d}\psi_{e}\rangle_{c}\simeq 0$, (62)
and (64) lead to a consistency equation for the two-point functions,

$\displaystyle
K_{ac}\langle\bar{\psi}_{b}\phi_{c}\rangle+4gB_{acde}\langle\bar{\psi}_{b}\psi_{e}\rangle\langle\bar{\psi}_{c}\psi_{d}\rangle=\delta_{ab}.$
(65)

In fact, the equation (65) can be written in another way. Let us denote
$G_{ab}=\langle\bar{\psi}_{a}\psi_{b}\rangle$. Then, by applying $G^{-1}$ on
both sides of (65), we obtain

$\displaystyle K_{ab}+4gB_{acdb}G_{cd}=G^{-1}_{ba}.$ (66)

By solving this equation one can determine $G$.

Once we know $G$, the partition function $Z$ can be computed in the following
manner. We first note that

$\displaystyle\begin{split}\frac{d}{dg}\log
Z&=B_{abcd}\langle\bar{\psi}_{a}\bar{\psi}_{b}\psi_{c}\psi_{d}\rangle\\\
&=2B_{abcd}G_{ad}G_{bc}\end{split}$ (67)

with the approximation taken above. Therefore one can determine $Z$ by solving
the equation (67) with the initial condition,

$\displaystyle Z(g=0)=\det K.$ (68)

The procedure above can be summarized as a more intuitive process. Let us
define

$\displaystyle\begin{split}S_{\rm eff}&=\langle S\rangle-\log\det G\\\
&=K_{ab}G_{ab}+2gB_{abcd}G_{ad}G_{bc}-\log\det G,\end{split}$ (69)

where $\langle S\rangle$ in (69) has been computed with the approximation
taken above. Here it may be more appropriate to replace $\log\det G$ with
$\log(-\det(G))$, depending on the sign of $\det G$ of a solution. In (38) it
is chosen to be the latter because the solution turns out to give $\det G<0$.
Anyway, the difference is a constant and hence does not essentially affect the
discussions below.

Then we consider the stationary condition with respect to $G$:

$\displaystyle\begin{split}0&=\frac{\partial}{\partial G_{ab}}S_{\rm eff}\\\
&=K_{ab}+4gB_{acdb}G_{cd}-G^{-1}_{ba}.\end{split}$ (70)

This indeed agrees with (66).

Moreover, let us define

$\displaystyle Z^{0}_{\rm eff}=\det K\,e^{S^{0}_{\rm eff}-S_{\rm
eff}^{0}(g=0)},$ (71)

where $S^{0}_{\rm eff}$ is defined by inserting the solution $G=G^{0}$ of (70)
(or (66)) into (69). Then we can find that

$\displaystyle\frac{d}{dg}\log Z^{0}_{\rm eff}=\frac{\partial S^{0}_{\rm
eff}}{\partial g}+\frac{\partial S^{0}_{\rm eff}}{\partial
G^{0}_{ab}}\frac{dG^{0}_{ab}}{dg}=2gB_{abcd}G^{0}_{ad}G^{0}_{bc},$ (72)

where we have used the stationary condition (70). Indeed (72) agrees with
(67). We also find $Z^{0}_{\rm eff}(g=0)=\det K$. Therefore, $Z^{0}_{\rm eff}$
gives the partition function in this approximation.

##  Appendix Appendix B The solution to (40)

The solution to (40) reproducing (41) at $v=0$ is given by

$\displaystyle\begin{split}&Q_{11}=\frac{-2+\sqrt{2}\sqrt{\frac{-1+\epsilon+4x+\sqrt{(1+\epsilon-4x)^{2}+16\epsilon
x}}{\epsilon}}}{4x},\\\ &Q_{12}=Q_{21}\\\ &\ \ \ \ \
=\frac{1}{8x}\left(-4+\sqrt{2}\sqrt{\frac{-1+\epsilon+4x+\sqrt{(1+\epsilon-4x)^{2}+16\epsilon
x}}{\epsilon}}\right.\\\ &\hskip
56.9055pt-\sqrt{2}\epsilon\sqrt{\frac{-1+\epsilon+4x+\sqrt{(1+\epsilon-4x)^{2}+16\epsilon
x}}{\epsilon}}\\\ &\hskip
56.9055pt-4\sqrt{2}x\sqrt{\frac{-1+\epsilon+4x+\sqrt{(1+\epsilon-4x)^{2}+16\epsilon
x}}{\epsilon}}\\\ &\hskip
56.9055pt\left.+\sqrt{2}\sqrt{(1+\epsilon-4x)^{2}+16\epsilon
x}\sqrt{\frac{-1+\epsilon+4x+\sqrt{(1+\epsilon-4x)^{2}+16\epsilon
x}}{\epsilon}}\right),\\\
&Q_{22}=-\frac{\epsilon\left(-2+\sqrt{2}\sqrt{\frac{-1+\epsilon+4x+\sqrt{(1+\epsilon-4x)^{2}+16\epsilon
x}}{\epsilon}}\right)}{4x}.\end{split}$ (73)

## References

  * [1] E. Wigner, “On the Distribution of the Roots of Certain Symmetric Matrices”, Annals of Mathematics 67 (2), 325-327 (1958) https://doi.org/10.2307/1970008.
  * [2] D. J. Gross and E. Witten, “Possible Third Order Phase Transition in the Large N Lattice Gauge Theory,” Phys. Rev. D 21, 446-453 (1980) doi:10.1103/PhysRevD.21.446
  * [3] S. R. Wadia, “$N$ = Infinity Phase Transition in a Class of Exactly Soluble Model Lattice Gauge Theories,” Phys. Lett. B 93, 403-410 (1980) doi:10.1016/0370-2693(80)90353-6
  * [4] E. Brezin, C. Itzykson, G. Parisi and J. B. Zuber, “Planar Diagrams,” Commun. Math. Phys. 59, 35 (1978) doi:10.1007/BF01614153
  * [5] J. Ambjorn, B. Durhuus and T. Jonsson, “Three-dimensional simplicial quantum gravity and generalized matrix models,” Mod. Phys. Lett. A 6, 1133-1146 (1991) doi:10.1142/S0217732391001184
  * [6] N. Sasakura, “Tensor model for gravity and orientability of manifold,” Mod. Phys. Lett. A 6, 2613-2624 (1991) doi:10.1142/S0217732391003055
  * [7] N. Godfrey and M. Gross, “Simplicial quantum gravity in more than two-dimensions,” Phys. Rev. D 43, 1749-1753 (1991) doi:10.1103/PhysRevD.43.R1749
  * [8] R. Gurau, “Colored Group Field Theory,” Commun. Math. Phys. 304, 69-93 (2011) doi:10.1007/s00220-011-1226-9 [arXiv:0907.2582 [hep-th]].
  * [9] L. Qi, H. Chen, Y, Chen, “Tensor Eigenvalues and Their Applications”, Springer, Singapore, 2018.
  * [10] M. Ouerfelli, V. Rivasseau and M. Tamaazousti, “The Tensor Track VII: From Quantum Gravity to Artificial Intelligence,” [arXiv:2205.10326 [hep-th]].
  * [11] L. Qi, “Eigenvalues of a real supersymmetric tensor,” Journal of Symbolic Computation 40, 1302-1324 (2005).
  * [12] L.H. Lim, “Singular Values and Eigenvalues of Tensors: A Variational Approach,” in Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP ’05), Vol. 1 (2005), pp. 129–132.
  * [13] D. Cartwright and B. Sturmfels, “The number of eigenvalues of a tensor,” Linear algebra and its applications 438, 942-952 (2013).
  * [14] P. Breiding, “The expected number of eigenvalues of a real gaussian tensor,” SIAM Journal on Applied Algebra and Geometry 1, 254-271 (2017).
  * [15] P. Breiding, “How many eigenvalues of a random symmetric tensor are real?,” Transactions of the American Mathematical Society 372, 7857-7887 (2019).
  * [16] O. Evnin, “Melonic dominance and the largest eigenvalue of a large random tensor,” Lett. Math. Phys. 111, 66 (2021) doi:10.1007/s11005-021-01407-z [arXiv:2003.11220 [math-ph]].
  * [17] R. Gurau, “On the generalization of the Wigner semicircle law to real symmetric tensors,” [arXiv:2004.02660 [math-ph]].
  * [18] N. Sasakura, “Signed distributions of real tensor eigenvectors of Gaussian tensor model via a four-fermi theory,” [arXiv:2208.08837 [hep-th]].
  * [19] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, (Clarendon Press, Oxford, 1989).
  * [20] M. Headrick, J Michelson, grassmann.m, https://people.brandeis.edu/~headrick/Mathematica/.
  * [21] T. Kawano and N. Sasakura, “Emergence of Lie group symmetric classical spacetimes in the canonical tensor model,” PTEP 2022, no.4, 043A01 (2022) doi:10.1093/ptep/ptac045 [arXiv:2109.09896 [hep-th]].
  * [22] N. Sasakura, “Splitting-merging transitions in a tensor-vectors system in exact large-$N$ limits,” [arXiv:2206.12017 [hep-th]].


# A Note on Sun’s Conjectures on Apéry-like Sums Involving Lucas Sequences and
Harmonic Numbers

Ce Xua, and Jianqiang Zhaob,
a. School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002,
P.R. China
b. Department of Mathematics, The Bishop’s School, La Jolla, CA 92037, USA
Email<EMAIL_ADDRESS>first author, ORCID 0000-0002-0059-7420.Email:
<EMAIL_ADDRESS>corresponding author, ORCID 0000-0003-1407-4230.

Abstract. In this paper, we will prove Zhi-Wei Sun’s four conjectural
identities on Apéry-like sums involving Lucas sequences and harmonic numbers
by using a few results of Davydychev–Kalmykov [1].

Keywords: Apéry-like sums, harmonic numbers, Lucas numbers, Lucas sequences,
Clausen function.

AMS Subject Classifications (2020): 11M06, 11B39, 40A05

##  1 Introduction

The classical Lucas numbers $\\{L_{n}\\}_{n\geq 0}$ are defined by

$\displaystyle L_{0}=2,\ L_{1}=1\quad\text{and}\quad L_{n+1}=L_{n}+L_{n-1}\
\forall n\geq 1.$

Obviously, for any $n\geq 0$,

$L_{n}=\Bigl{(}\displaystyle\frac{1+\sqrt{5}}{2}\Bigr{)}^{n}+\Bigl{(}\displaystyle\frac{1-\sqrt{5}}{2}\Bigr{)}^{n}.$

Let $A$ and $B$ be integers. Similarly to Sun [3], we define the more general
Lucas sequence $v_{n}=v_{n}(A,B)$ as follows:

$\displaystyle v_{0}(A,B):=2,\quad v_{1}(A,B):=A\quad\text{and}\quad
v_{n+1}(A,B)=Av_{n}(A,B)-Bv_{n-1}(A,B)\ \forall n\geq 1.$

The characteristic equation $x^{2}-Ax+B=0$ of the sequence $\\{v_{n}\\}_{n\geq
0}$ has two roots

$\alpha=\displaystyle\frac{A+\sqrt{A^{2}-4B}}{2}\quad\text{and}\quad\beta=\displaystyle\frac{A-\sqrt{A^{2}-4B}}{2}.$

It is well known that for any integer $n\geq 0$ we have

$\displaystyle v_{n}(A,B)=\alpha^{n}+\beta^{n}.$

For examples, setting $\phi=\displaystyle\frac{\sqrt{5}+1}{2}$ we have

$\displaystyle v_{n}(3,1)=L_{2n}=\phi^{2n}+\phi^{-2n},$ $\displaystyle
v_{n}(4,1)=(2-\sqrt{3})^{n}+(2+\sqrt{3})^{n},$ $\displaystyle
v_{n}(4,2)=(2-\sqrt{2})^{n}+(2+\sqrt{2})^{n},$ $\displaystyle
v_{n}(5,5)=(\sqrt{5}\phi)^{n}+(\sqrt{5}\phi^{-1})^{n}.$

Define $H_{0}:=0$ and the classical harmonic numbers

$H_{n}:=\displaystyle\sum_{k=1}^{n}\displaystyle\frac{1}{k}\quad\forall n\geq
1.$

In this short note, we will prove the following results.

######  Theorem 1.1.

We have

$\displaystyle\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{L_{2n}}{n^{2}\binom{2n}{n}}\left(H_{2n}-H_{n-1}\right)=\displaystyle\frac{41\zeta(3)+4\pi^{2}\log(\phi)}{25},$
(1.1)
$\displaystyle\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{v_{n}(5,5)}{n^{2}\binom{2n}{n}}\left(H_{2n}-H_{n-1}\right)=\displaystyle\frac{124\zeta(3)+\pi^{2}\log\left(5^{5}\phi^{6}\right)}{50},$
(1.2)
$\displaystyle\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{v_{n}(4,1)}{n^{2}\binom{2n}{n}}\left(H_{2n}-H_{n-1}\right)=\displaystyle\frac{23\zeta(3)+2\pi^{2}\log(2+\sqrt{3})}{12},$
(1.3)
$\displaystyle\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{v_{n}(4,2)}{n^{2}\binom{2n}{n}}\left(H_{2n}-H_{n-1}\right)=\displaystyle\frac{259\zeta(3)+2\pi^{2}(4\log(2)+8\log(1+\sqrt{2}))}{128}.$
(1.4)

Motivated by his study of the congruences in [3], Sun [4, 5, 6] proposes
identities (1.1) and (1.2) as [5, Conjectures 10.62 and 10.63] (see also [4,
Eqs. (3.11) and (3.12)]). He further conjectures (1.3) and (1.4) in [6].

##  2 Proof of Theorem 1.1

Recall that for any positive integer $m$ the Clausen function ${\rm
Cl}_{m}(\theta)$ is defined by

$\displaystyle{{\rm
Cl}}_{2m-1}(\theta):=\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{\cos(n\theta)}{n^{2m-1}}\quad\text{and}\quad{{\rm
Cl}}_{2m}(\theta):=\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{\sin(n\theta)}{n^{2m}}\qquad\forall\theta\in[0,\pi].$

Many special values of the Clausen function are closely related to the Riemann
zeta values. For example,

${\rm Cl}_{2m-1}(0)=\zeta(2m-1),\quad{\rm
Cl}_{2m-1}(\pi)=(2^{2-2m}-1)\zeta(2m-1),\quad{\rm
Cl}_{2m-1}\Big{(}\displaystyle\frac{\pi}{2}\Big{)}=\displaystyle\frac{{\rm
Cl}_{2m-1}(\pi)}{2^{2m-1}}.$ (2.5)

The following relations among special values of the Clausen function are well
known (see, e.g., [2, Eq. (6.51)]): for any positive integer $r$ we have

$\displaystyle{\rm Cl}_{3}\big{(}\tfrac{\pi}{r}\big{)}+{\rm
Cl}_{3}\big{(}\tfrac{3\pi}{r}\big{)}+\cdots+{\rm
Cl}_{3}\big{(}\tfrac{(2r-1)\pi}{r}\big{)}=-\tfrac{3}{4r^{2}}\zeta(3),$ (2.6)
$\displaystyle{\rm Cl}_{3}\big{(}\tfrac{2\pi}{r}\big{)}+{\rm
Cl}_{3}\big{(}\tfrac{4\pi}{r}\big{)}+\cdots+{\rm
Cl}_{3}\big{(}\tfrac{(2r-2)\pi}{r}\big{)}=-\big{(}1-\tfrac{1}{r^{2}}\big{)}\zeta(3).$
(2.7)

Setting $r=4,5,6$ in (2.6) and $r=5$ in (2.7) one obtains

$\displaystyle{\rm Cl}_{3}\big{(}\tfrac{\pi}{4}\big{)}+{\rm
Cl}_{3}\big{(}\tfrac{3\pi}{4}\big{)}=-\tfrac{3}{128}\zeta(3),\qquad{\rm
Cl}_{3}\big{(}\tfrac{\pi}{5}\big{)}+{\rm
Cl}_{3}\big{(}\tfrac{3\pi}{5}\big{)}=\tfrac{9}{25}\zeta(3),$ (2.8)
$\displaystyle{\rm Cl}_{3}\big{(}\tfrac{\pi}{6}\big{)}+{\rm
Cl}_{3}\big{(}\tfrac{5\pi}{6}\big{)}=\tfrac{1}{12}\zeta(3),\qquad{\rm
Cl}_{3}\big{(}\tfrac{2\pi}{5}\big{)}+{\rm
Cl}_{3}\big{(}\tfrac{4\pi}{5}\big{)}=-\tfrac{12}{25}\zeta(3),$ (2.9)

since ${\rm Cl}_{3}(\theta)={\rm Cl}_{3}(2\pi\pm\theta)$, ${\rm
Cl}_{3}(\pi)=-\tfrac{3}{4}\zeta(3)$ and ${\rm
Cl}_{3}\big{(}\tfrac{\pi}{2}\big{)}=-\tfrac{3}{32}\zeta(3)$ by (2.5).

On the other hand, Davydychev and Kalmykov have shown that (see [1, (2.36),
(2.37) and (2.67)])

$\displaystyle\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{1}{n^{3}\binom{2n}{n}}u^{n}=2{\rm
Cl}_{3}(\theta)+2\theta{\rm
Cl}_{2}(\theta)-2\zeta(3)+\theta^{2}\log\Big{(}2\sin\displaystyle\frac{\theta}{2}\Big{)},$
$\displaystyle\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{H_{n-1}}{n^{2}\binom{2n}{n}}u^{n}=4{\rm
Cl}_{3}(\pi-\theta)-2\theta{\rm Cl}_{2}(\pi-\theta)+3\zeta(3),$
$\displaystyle\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{H_{2n-1}}{n^{2}\binom{2n}{n}}u^{n}=-2{\rm
Cl}_{3}(\theta)+4{\rm Cl}_{3}(\pi-\theta)-2\theta{\rm
Cl}_{2}(\pi-\theta)-\theta{\rm Cl}_{2}(\theta)+5\zeta(3),$

where $u=4\sin^{2}\displaystyle\frac{\theta}{2}$ for any $\theta\in[0,\pi]$.
Combining the above three equations we get

$\displaystyle\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{u^{n}}{n^{2}\binom{2n}{n}}\left(H_{2n}-H_{n-1}\right)$
$\displaystyle=\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{u^{n}}{n^{2}\binom{2n}{n}}\left(H_{2n-1}-H_{n-1}+\displaystyle\frac{1}{2n}\right)$
$\displaystyle=\zeta(3)-{\rm
Cl}_{3}(\theta)+\displaystyle\frac{\theta^{2}}{2}\log\Big{(}2\sin\displaystyle\frac{\theta}{2}\Big{)}.$
(2.10)

Taking special values of $u$ we obtain

$\displaystyle u=\Big{(}\displaystyle\frac{\sqrt{5}\pm
1}{2}\Big{)}^{2}=4\sin^{2}\displaystyle\frac{\theta}{2}\quad\Longrightarrow\quad$
$\displaystyle\sin\displaystyle\frac{\theta}{2}=\displaystyle\frac{\sqrt{5}\pm
1}{4}\quad\text{and}\quad\theta=\displaystyle\frac{2\pi}{5}\pm\displaystyle\frac{\pi}{5},$
$\displaystyle
u=\displaystyle\frac{5\pm\sqrt{5}}{2}=4\sin^{2}\displaystyle\frac{\theta}{2}\quad\Longrightarrow\quad$
$\displaystyle\sin\displaystyle\frac{\theta}{2}=\sqrt{\displaystyle\frac{5\pm\sqrt{5}}{8}}\quad\text{and}\quad\theta=\displaystyle\frac{3\pi}{5}\pm\displaystyle\frac{\pi}{5},$
$\displaystyle
u=2\pm\sqrt{3}=4\sin^{2}\displaystyle\frac{\theta}{2}\quad\Longrightarrow\quad$
$\displaystyle\sin\displaystyle\frac{\theta}{2}=\displaystyle\frac{\sqrt{2\pm\sqrt{3}}}{2}\quad\text{and}\quad\theta=\displaystyle\frac{\pi}{2}\pm\displaystyle\frac{\pi}{3},$
$\displaystyle
u=2\pm\sqrt{2}=4\sin^{2}\displaystyle\frac{\theta}{2}\quad\Longrightarrow\quad$
$\displaystyle\sin\displaystyle\frac{\theta}{2}=\displaystyle\frac{\sqrt{2\pm\sqrt{2}}}{2}\quad\text{and}\quad\theta=\displaystyle\frac{\pi}{2}\pm\displaystyle\frac{\pi}{4}.$

Therefore, (2) and (2.8)–(2.9) yield

$\displaystyle\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{L_{2n}}{n^{2}\binom{2n}{n}}\left(H_{2n}-H_{n-1}\right)$
$\displaystyle=2\zeta(3)-{\rm Cl}_{3}\big{(}\tfrac{3\pi}{5}\big{)}-{\rm
Cl}_{3}\big{(}\tfrac{\pi}{5}\big{)}+\displaystyle\frac{4\pi^{2}}{25}\log(\phi)$
$\displaystyle=\displaystyle\frac{41\zeta(3)+4\pi^{2}\log(\phi)}{25},$
$\displaystyle\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{v_{n}(5,5)}{n^{2}\binom{2n}{n}}\left(H_{2n}-H_{n-1}\right)$
$\displaystyle=2\zeta(3)-{\rm Cl}_{3}\big{(}\tfrac{2\pi}{5}\big{)}-{\rm
Cl}_{3}\big{(}\tfrac{4\pi}{5}\big{)}+\displaystyle\frac{\pi^{2}\log\left(5^{5}\phi^{6}\right)}{50}$
$\displaystyle=\displaystyle\frac{124\zeta(3)+\pi^{2}\log\left(5^{5}\phi^{6}\right)}{50},$
$\displaystyle\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{v_{n}(4,1)}{n^{2}\binom{2n}{n}}\left(H_{2n}-H_{n-1}\right)$
$\displaystyle=2\zeta(3)-{\rm Cl}_{3}\big{(}\tfrac{\pi}{6}\big{)}-{\rm
Cl}_{3}\big{(}\tfrac{5\pi}{6}\big{)}+\displaystyle\frac{\pi^{2}\log(2+\sqrt{3})}{6}$
$\displaystyle=\displaystyle\frac{23\zeta(3)+2\pi^{2}\log(2+\sqrt{3})}{12},$
$\displaystyle\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{v_{n}(4,2)}{n^{2}\binom{2n}{n}}\left(H_{2n}-H_{n-1}\right)$
$\displaystyle=2\zeta(3)-{\rm Cl}_{3}\big{(}\tfrac{\pi}{4}\big{)}-{\rm
Cl}_{3}\big{(}\tfrac{3\pi}{4}\big{)}+\displaystyle\frac{\pi^{2}(4\log(2)+8\log(1+\sqrt{2}))}{64}$
$\displaystyle=\displaystyle\frac{259\zeta(3)+2\pi^{2}(4\log(2)+8\log(1+\sqrt{2}))}{128}.$

This concludes the proof of Theorem 1.1 and this note.

Acknowledgement. Ce Xu expresses his deep gratitude to Prof. Zhi-Wei Sun for
valuable discussions and comments. Ce Xu is supported by the National Natural
Science Foundation of China (Grant No. 12101008), the Natural Science
Foundation of Anhui Province (Grant No. 2108085QA01) and the University
Natural Science Research Project of Anhui Province (Grant No. KJ2020A0057).
Jianqiang Zhao is supported by the Jacobs Prize from The Bishop’s School.

## References

  * [1] A.I. Davydychev and M. Yu. Kalmykov, Massive Feynman diagrams and inverse binomial sums, _Nuclear Phys. B_ 699 (2004), pp. 3–64. arXiv:hep-th/0303162v4.
  * [2] L. Lewin, _Polylogarithms and Associated Functions_ , Elsevier Sci. Publishers, New York, New York, 1981.
  * [3] Z.-W. Sun, Curious congruences for Fibonacci numbers, arXiv:0912.2671.
  * [4] Z.-W. Sun, New series for some special values of $L$-functions, _Nanjing Univ. J. Math. Biquarterly_ 32(2015), no.2, 189-218.
  * [5] Z.-W. Sun, _New Conjectures in Number Theory and Combinatorics_ , Harbin Institute of Technology Press, (in chinese) 2021.
  * [6] Z.-W. Sun, Evaluations of two new series involving Lucas $v$-sequences, Question 420049 at MathOverflow, April 9, 2022. https://mathoverflow.net/questions/420049.