数理解析研究所講究録
第 1911 巻 2014 年 130-149

130

Problems on Low-dimensional Topology, 2014
Edited by T. Ohtsuki1

This is a list of open problems on low-dimensional topology with expositions of
their history, background, significance, or importance. This list was made by editing
manuscripts written by contributors of open problems to the problem session of theconference “Intelligence of Low-dimensional Topology”’ held at Research Institutefor Mathematical Sciences, Kyoto University in May 21-23, 2014.

Contents
1 Applications of Laver tables to braids

2

2 Similarity between number theory and knot theory

4

3 Iwasawa invariants of cyclic covers of link exteriors

44 Profinite knots

6

5 Invariants of knots derived from the algebraic $K$ -theory7

6 1-cocycles in the space of knots

8

7 Canonical arc index of cable links

9

8 The state numbers for virtual knots

11

9 Local and global properties of graphs

13

10 Essential tribranched surfaces in 3-manifolds

14

11 Invariants of homology 3-spheres motivated by the Chern-Simons
16
perturbation theory

lResearch Institute for Mathematical Sciences, Kyoto University, Sakyo-ku, Kyoto, 606-8502, JAPAN
Email: tomotada@kurims.kyoto-u.ac.jp

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1

Applications of Laver tables to braids(Victoria Lebed)

A Laver table
is the set 1, 2, 3, . . . ,endowed with the unique binary
operationsatisfying the (lefl) self-distributivity condition$A_{n}$$\{$$2^{n}\}$$\triangleright_{n}$(1)$a\triangleright_{n}(b\triangleright n^{\mathcal{C})}=(a\triangleright_{n}b)\triangleright_{n}(a\triangleright n^{\mathcal{C})}$and the initial condition $a\triangleright_{n}1\equiv a+1(mod 2^{n})$ . They form an infinite familyof finite shelves ( sets endowed with a self-distributive operation). The smallest
examples are presented in Figure 1; the cell $(i,j)$ of the table forcontains
In the table for
one recognizes operation “implication” from Logic.
$=$$A_{n}$$i\triangleright_{n}j.$$A_{1}$

Figure 1: Multiplication tables for the first four Laver tables

Laver tables were discovered by Richard Laver as a by-product of his analysis of
iterations of elementary embeddings in Set Theory [33]. Since then this structure
was redefined in elementary terms (as above), and many of its deep properties were
given a combinatorial proof. However, as for now, some fundamental facts about
Laver tables have been established only under an unprovable large cardinal axiom.
One of them states that the inverse limit of thecontains a copy of the shelffreely generated by a single element.
Shelves have gained recognition among knot theorists due to coloring techniques.
Namely, a coloring of a positive braid diagram $D$ by a shelf
assigns an elementof to every arc of $D$ in such a way that a -colored strand becomes-coloredwhen it over-crosses an -colored strand. Figure 2 shows that this coloring condition
is compatible with the RIII move thanks to (1). Hence the number of -coloringsof diagrams yields an invariant of underlying positive braids. This invariant extends
to arbitrary braids if
is a rack $(i.e.$ , for all , the map
is bijective
on $Q)$ , and to knots if is a quandle $(=a$ rack where every element is idempotent:. Such -coloring counting invariants turn out to be extremely powerful
and well adapted for actual calculations.
Now, even though the free shelf
is not a rack, the -colorings of arbitrary
braid diagrams are still meaningful. In particular, when applied to braid diagrams$A_{n}$$\mathcal{F}_{1}$$(Q, \triangleright)$

$Q$

$b$

$(a\triangleright b)$

$a$

$Q$

$Q$

$b$

$a\mapsto b\triangleright a$$Q$$a\triangleright a=a)$$Q$$\mathcal{F}_{1}$$\mathcal{F}_{1}$

2

132

$\sim$

Figure 2: Reidemeister III move

-distributivity$\Leftrightarrow self$

presented in the normal form, they lead to a construction of a well-behaved orderon braids [12]. Since conjecturally Laver tables are finite approximations of , it$\mathcal{F}_{1}$is natural to expect that-colorings can also say a lot about arbitrary braids.
Moreover, because of the finiteness, they are well adapted for computations. The
following question thus seems very promising:
Question 1.1 (P. Dehornoy [13]). How can one extract topological information fromthe colorings by Laver tables in the case of arbitrary braids
A possible strategy for answering this question involves rack cohomology, as developed in [17, 8]. For a shelf, rack cohomology
is defined as thecohomology of the complex, where,$A_{n}$$H_{R}^{k}(Q)$$(Q, \triangleright)$$d_{R}^{k}= \sum_{i=1}^{k+1}(-1)^{i-1}(d_{i}^{k}-\tilde{d}_{i}^{k})$$(Hom(Q^{\cross k}, \mathbb{Z}), d_{R}^{k})$and$(d_{i}^{k}f)(a_{1}, \ldots , a_{k+1})=f(a_{1}, \ldots, a_{i-1}, a_{i}\triangleright a_{i+1}, .

.

.

, a_{i}\triangleright a_{k+1})$$(\tilde{d}_{i}^{k}f)(a_{1}, \ldots, a_{k+1})=f(a_{1}, \ldots, a_{i-1}, a_{i+1},\ldots, a_{k+1})$

The 2-cocycles from this cohomology theory
satisfying

–

,

.

that is, maps$\phi$:$Q\cross Qarrow \mathbb{Z}$$\phi(a\triangleright b, a\triangleright c)+\phi(a, c)=\phi(a, b\triangleright c)+\phi(b, c)$–are of particular importance. Evaluate such a 2-cocycle on the colors adjacent
to each crossing of a -colored positive braid diagram as shown on Figure 3, andsum up the values obtained. Figure 3 proves the multi-set of the results of thissummation for all possible -colorings to be an invariant of positive braids. These
$Q$

$Q$

-coloring cocycle invariants sharpen the -coloring counting invariants: the latterare obtained by considering any constant 2-cocycle . A slight modification of thismethod involves region coloring and rack 3-cocycles.
$Q$

$Q$

$\phi$$(j((1, b)+(^{1}\prime)^{:}(a, c)+(\prime\supset^{:}(a\triangleright b, 0\triangleright c)$

Figure 3: Two-cocycle$\phiarrow$diagram weights. They form
In [14], we explicitly described rack 2- and 3-cocycles for all the$2^{2n}-2^{n}+1$, giving rich families
and, respectively,
free modules over of rank
$A_{n}$$\mathbb{Z}$$2^{n}$3

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of -coloring cocycle invariants. We showed that these cocycles capture important
combinatorial properties of the. Question 1.1 can now be narrowed as follows:$A_{n}$$A_{n}$

Question 1.2 (Dehornoy-Lebed [14]). How can one extract topological informationfrom the colorings by Laver tables, weighted by rack 2- or 3-cocycles, in the case ofarbitrary braids./?

Independently of topological applications, rack cohomology calculations for Lavertables can be instrumental for a better understanding of the structure of theIn [14], we established thatfor all and for $k\leq 3.$$A_{n}.$$H_{R}^{k}(A_{n})\simeq \mathbb{Z}$$n$Conjecture 1.3 (V. Lebed). For all Laver tables
and integers , the rack kcocycles for, whereis a degree $k-1$
form free modules over of rankpolynomial with integer coeﬃcients. Moreover, one has, with (theequivalence class of) the constant cocycle
as generator.
$k$$A_{n}$$A_{n}$$\mathbb{Z}$$\theta_{k}(2^{n})$$\theta_{k}$$H_{R}^{k}(A_{n})\simeq \mathbb{Z}$$\phi(a_{1}, .

.

.

, a_{k})=1$

Preliminary computations confirm this conjecture for $k=4$ . However, calculation
methods for general are still missing. It would also be interesting to find explicit
formulas for the polynomials and to study their properties.
Further, as follows from the work of A. Dr\’apal [15, 55], all finite shelves witha single generator can be regarded as “interpolations” between Laver tables andcyclic shelves $C_{m}(i.e.$ , sets 1, 2, 3,
with the operation defined by$b+1$ (modm)). Like for Laver tables, first cohomology groups for theturn out
to be isomorphic to$k$$\theta_{k}$$\{$$\ldots,$

$m\}$

$a$

$o_{m}b\equiv$$C_{m}$$\mathbb{Z}.$Conjecture 1.4 (V. Lebed). For all finite mono-generated shelves$Q$, one has

$H_{R}^{k}(Q)\simeq \mathbb{Z}.$2

Similarity between number theory and knot theory(Teruhisa Kadokami)

Number Theory and Knot Theory are said ‘similar’ because Galois Theory in
Number Theory and Alexander Theory in Knot Theory are similar theoretically[39, 43]. They may be unified from a group theoretical view point; see Figure 4.

Problem 2.1. Describe clearly a unified notion of Number Theory and Knot Theory.

3

Iwasawa invariants of cyclic covers of link exteriors(Yasushi Mizusawa)

Let be a link in a rational homology 3-sphere $M$ , and let $X$ be the exterior ofwith the fundamental group $G_{L}=\pi_{1}(X)$ . A surjective homomorphism :corresponds to an infinite cyclic cover
over $X$ . Let
be the subcover of degree, for a fixed prime number . Then we obtain a towerof cyclic branchedcoversof $M$ as the Fox completions of. Based on the analogy between
$L$

$L$

$\sigma$$X_{\sigma}$$p^{n}$$X_{\sigma,p^{n}}$$\{M_{\sigma,p^{n}}\}_{n}$$p$$M_{\sigma,p^{n}}$$X_{\sigma,p^{n}}$4$G_{L}arrow \mathbb{Z}$134$\cup$

groupssome structures

palrs

$0$

$W1$

Iwasawa module
Iwasawapolynomial

A exander module

Alexanderpolynomial

Figure 4: Similarity between Number Theory and Knot Theory

knot theory and number theory, Morishita . al. ([25, 28, 42]) gave an analogue of
Iwasawa’s class number formula (cf. [61] etc Assume thatis finite
for all $n\geq$ O. Then there are non-negative integers
and an integer(possibly negative) such that$et$$H_{1}(M_{\sigma,p^{n}};\mathbb{Z})$$\lambda_{L,\sigma},$$\nu_{L,\sigma}$$\mu_{L,\sigma}$$v_{p}(\# H_{1}(M_{\sigma,p^{n}};\mathbb{Z}))=\lambda_{L,\sigma}n+\mu_{L,\sigma}p^{n}+\nu_{L,\sigma}$

for all suﬃciently large , where
denotes the -adic valuation normalized as$v_{p}(p)=1$ . These invariants
are called Iwasawa invariants.
Professor Masato Kurihara gave us the following problem.
$n$

$p$

$v_{p}$$\lambda_{L,\sigma},$$\mu_{L,\sigma},$$\nu_{L,\sigma}$

Problem 3.1. Refine the formula removing the finiteness ofa -adic growth formula for the torsion part of.

$H_{1}(M_{\sigma,p^{n}};\mathbb{Z})$, i. e., give$H_{1}(M_{\sigma,p^{n}};\mathbb{Z})$$p$

The following problem also seems to be considerable. A partial result is obtainedin [29].

Problem 3.2 (T. Kadokami, Y. Mizusawa). Determine the possible values of
and$\mu_{L,\sigma}$$\lambda_{L,\sigma},$$v_{L,\sigma}.$

We often consider a non-archimedean prime (a prime ideal of the integer ringof a number field) as an analogue of 1-component link (a knot). However, in theanalogies between Alexander-Fox theory and Iwasawa theory, the set of all prime
ideals lying over a fixed prime number in a totally real number field looks like 1component link. It seems that the following problem is considerable to understandthe archimedean primes more deeply in arithmetic topology.
$p$

5

135
Problem 3.3 (T. Kadokami, Y. Mizusawa). What is “the number of components”’
of a prime$/p$is an -component link in , the link moduleand the Alexander module
are modules over
. Then there is a natural injective-homomorphism : $B_{L}\mapsto A_{L}$ . Let
generatedbe the -submodule ofby meridional elements (cf. [24]). Motivated by Greenberg’s conjecture (cf. [21]) in
Iwasawa theory, the following problem is proposed (with some examples) in [29],which is not a strict analogue of the conjecture.

If

$L$

$S^{3}$$r$$A_{L}$$B_{L}$$\Lambda=\mathbb{Z}[t_{1}^{\pm 1}, \cdots, t_{r}^{\pm 1}]$$\Lambda$$\theta$$A_{L}$$\Lambda$$M_{L}$

Problem 3.4 (T. Kadokami, Y. Mizusawa). When is $Y_{L}:=B_{L}/\theta^{-1}(M_{L})$ a pseudonull-module?

$\Lambda$

If we have a homomorphism :with large image, we obtain a padic Lie tower over X. (The tower
is a -adic Lie tower for $d=1.$ ) In [3, 7]
etc., the growth of Betti numbers in a -adic Lie tower is studied. The invariant
can be regarded as a kind of Betti numbers. Motivated by analogous studies([50] etc.) in noncommutative Iwasawa theory, we have the following problem.
$\rho$$G_{L}arrow GL_{d}(\mathbb{Z}_{p})$$\{X_{\sigma,p^{n}}\}_{n}$

$p$

$p$

$\lambda_{L,\sigma}$

Problem 3.5 (T. Kadokami, Y. Mizusawa). Give Iwasawa type formulas for -adic
Lie towers of branched covers over , with many examples.
$p$

$L$

4

Profinite knots(Hidekazu Furusho)Let
be the set of isotopy classes of oriented (topological) knots, which forms
a commutative monoid by the connected sum. Let
be the monoid of profinite
knots constructed in [18]. The set
forms a topological commutative monoid by
the connected sum and there is a natural monoid homomorphism$\mathcal{K}$$\hat{\mathcal{K}}$$\hat{\mathcal{K}}$$h:\mathcal{K}arrow\hat{\mathcal{K}}$whose image is dense in$\hat{\mathcal{K}}$, as is shown in [18].

Problem 4.1 (H. Furusho). Is the map$h$injectiv$e^{}?$

If it is non-injective, then the Kontsevich knot invariant fails to be perfect.
is residually finite,
As for Artin braid group $B_{n}(n\geq 2)$ , it is known thatnamely, the natural map$B_{n}$$B_{n}arrow\hat{B_{n}}$is injective.
Problem 4.2 (H. Furusho). Is there any Alexander-Markov-like theorem for profinite link$s^{}?$

One can find several proofs of Alexander-Markov’s theorem for topological links([4,58,60,64] etc). However they look heavily based on a certain finiteness property,
which we may not expect the validity for profinite links.
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136
Let Frac be the fractional group of , which forms a topological commutative
group. The action of the absolute Galois group$:=Ga1()$ of rationals on
Frac was constructed in [18].
$\hat{\mathcal{K}}$$\hat{\mathcal{K}}$$\mathbb{Q}$$G_{\mathbb{Q}}$$\hat{\mathcal{K}}$Problem 4.3 (H. Furusho). Is the

As for the braid groups, theby$Bely_{\dot{1}}$$G_{\mathbb{Q}}$$G_{\mathbb{Q}}$-action on Rac-action on$\hat{B_{n}}$?

$\hat{\mathcal{K}}faithful’$is known to be faithful for $n\geq 3$’s theorem [2].

Problem 4.4 (H. Furusho). Does there exist any (co)homology theow(or any
$X$
)
group
any
(pro-)variety
theoryandfundamental
defined over such that
(resp.
) carries a naturalis identified with-action and Rak(resp.
) so that our-action on Fkak
can be derived from the$H_{\star}$$\mathbb{Q}$$\pi_{1}^{\star}$$\hat{\mathcal{K}}$$H_{\star}(X_{\overline{\mathbb{Q}}})$$\pi_{1}^{\star}(X_{\overline{\mathbb{Q}}})$$H_{\star}(X_{\overline{\mathbb{Q}}})$$\pi_{1}^{\star}(X_{\overline{\mathbb{Q}}})$$G_{\mathbb{Q}}$$\hat{\mathcal{K}}$$G_{\mathbb{Q}}$-action there.

$\int p$$G_{\mathbb{Q}}$5

Invariants of knots derived from the algebraic $K$-theory(Takefumi Nosaka)

We start preliminarily recalling $K$ -theoretic results in knot theory. In general, theconcept of $K$ -groups often plays a role to uniformly understand several mathematicalphenomena. As examples in 3-dimensional knot theory, fixing a commutative field $F,$
-groups obtained fromvalues in-representations of link groups$L)arrow SL_{2}(F)$ are much studied so far. However, as far as I have looked over, I
had found no second $K$ -value with respect to such representations ; so, in [49], I
proposed such an invariant valued in “Milnor-Witt-group $K_{2}^{MW}(F)$ ” in a naturalway, where the preferred longitude is a key in construction. Without the details,
the $K$ -invariants are roughly summarized as follows:$K_{1^{-}},$$SL_{2}$$K_{3}$$\pi_{1}(S^{3}\backslash$$f$$K_{2}$

However, the work on the
mysterious.

$K_{2}$-value was two years ago; so there are many something

Problem 5.1 (T. Nosaka). Give some applications of theor to number theory.

$K_{2}$-value to knot theory

Problem 5.2 (T. Nosaka). Describe an arithmetic meaning of theexample, how about the hyperbolic holonomy of the figure-eight knot

$K_{2}$-values. For$4_{1}9$

Problem 5.3 (T. Nosaka). If $F$ is a global field, give an example of some representation
which takes non-trivial global information in $K_{2}^{MW}(F)$ .
Actually, in the paper [49], I dealt with some parts of $K_{2}^{MW}(F)$ only arising fromlocal fields.
$\pi_{1}(S^{3}\backslash L)arrow SL_{2}(F)$Problem 5.4 (T. Nosaka). If a hyperbolic holonomy
under some quadratic field $F$ , the relation between the hyperbolicity ($\pi_{1}(S^{3}\backslash K)arrow SL_{2}(F)$$e.g_{\rangle}$

7

is closed
volume)137
and arithmetic properties $(e.g., a$ special $value of the zeta$ function
is well studied(see a conprehensive book [37]). Discover an arithmetic relation of the-value.
$Furthermore_{f}$ compare the-value with Birsh-Tate conjecture that states theequality$\xi_{F})$$K_{2}$$K_{2}$$\zeta_{F}(-1)=?(-1)^{r_{1}}|K_{2}(\mathcal{O}_{F})|/|K_{3}(\mathcal{O}_{F})|,$

where $F$ is a totally real field, and$\mathcal{O}_{F}$is the ring of integers in $F.$

To solve them, the following two interpretations of $K_{2}^{MW}(F)$ in number theory would
be useful: First, from -homotopy theory, two isomorphisms$\mathbb{A}^{1}$$K_{2}^{MW}(F)\cong\pi_{1}^{A^{1}}(SL_{2}(F))\cong\pi_{1}^{A^{1}}(\mathbb{A}^{2}\backslash 0)$

hold via-Galois correspondence”’ or “‘(stable) sphere -spectrum” ; see [41] fordetails. Next, following Merkujev-Suslin theorem, we can analyse the $m$-torsion partof $K_{2}^{MW}(F)$ fromor the Brauer $m$-group $\pi Br(F)$ via “the Galois$F$
symbol if contains a primitive m-th root of unity; see, $e.g.$ , [63].
$\mathbb{A}^{1}$$\mathbb{A}^{1}$$H_{e’t}^{2}(Spec(F);\mu_{m}^{\otimes 2})$

Incidentally, it is not invaluable to consider some problems parallel to knottedsurfaces:
Problem 5.5 (T. Nosaka). With respect to a knotted surface
with a representation, construct invariants which are valued inor -group of $F.$
$K\subset S^{4}$$\pi_{1}(S^{4}\backslash K)arrow SL_{2}(F)$$K_{2;}K_{3}$$K_{4}$

6

1-cocycles in the space of knots(Arnaud Mortier)Following V.A.Vassiliev [59] we consider the space of smooth immersions
that coincide with the map $t\mapsto(O,0, t)$ outside of the segment [-1, 1]. The
subset of singular immersions (that are not embeddings) is denoted by and called
the discriminant of . The complement
is the set of (smooth) long knots.
When speaking of the cohomology of the space of knots, we mean the cohomologyof. At the zeroth level, the cohomology groupis the set ofvalued knot invariants. Our main object of interest here is the first cohomology
group, shortly denoted by. In [59], Vassiliev introduced particular
cohomology classes of the space of knots, well-known in degree as finite-type knot
invariants. Vassiliev (or finite-type) 1-cohomology classes form a subgroup. of$\mathbb{R}arrow$$\mathcal{K}$$\mathbb{R}^{3}$$\Sigma$$\mathcal{K}$$\mathcal{K}\backslash \Sigma$$\mathcal{K}\backslash \Sigma$$H^{1}(\mathcal{K}\backslash \Sigma;\mathbb{Q})$$H^{0}(\mathcal{K}\backslash \Sigma_{1\mathbb{Q})}$$\mathbb{Q}-$$H^{1}$$0$$H_{f.t}^{1}$$H^{1}.$

Question 6.1 (A. Mortier). Is there an axiomatic description of. similar to thewell-known axiomatization of Vassiliev invariants by J.Birman and X.-S. $Lin[5]’$?
$H_{f.t}^{1}$

One interest of considering 1-cocycles is that their evaluation on specific cycles
leads to knot invariants. For this, the cycles need to be defined independently of
the knot. Examples of such cycles are:$\bullet$rot (K) , the positive rotation of a long knot $K$ around its axis;
8

138$\bullet$$\bullet$drag (K) , the dragging of the second component of the connected sum $K\# K$
along the first (as in the proof of the equality $K\# L=L\# K$ );Hat (K) , the Hatcher cycle, which consists of dragging a “ball at infinity”’ allalong a fixed representation of a knot in$S^{3}.$

Problem 6.2 (A. Mortier). Find other examples of cycles in the space
are defined independently of the knot.

$\mathcal{K}\backslash \Sigma$, that

Question 6.3 (A. Mortier). The evaluation on the above cycles defines mapsdrag\star andfrom to the set of knot invariants . Is it true that under thesemaps, the image of a finite-type 1-cocycle is a finite-type invariant?
$rot^{\star},$$Hat^{\star}$$H^{1}$$H^{0}$

Since M.Polyak and O.Viro [51], followed by M.Goussarov [19], it is known that
finite-type knot invariants can be represented by means of linear combinations ofGauss diagrams. This idea was extended in [44] to produce combinatorial presentations of 1-cocycles, as follows.
We fix a linear projectionthat is generic with respect to the axis oflong knots. Then, given a cycle of knot diagrams, we make a list of the Reidemeister moves involved, depicted as Gauss diagrams with singularities, and count theirsubdiagrams with prescribed weights that define the 1-cocycle.
$\mathbb{R}^{3}arrow \mathbb{R}^{2}$

Question 6.4 (A. Mortier). Is it true that any 1-cocycle with a combinatorial description as above represents a finite-type cohomology class.
$(p$

For finite-type invariants there is a link between Polyak-Viro’s combinatorial
formulas and integral formulas, described in [52]. Integral formulas for 1-cocycles
have been found by K.Sakai in [53].
Question 6.5 (A. Mortier). Is there a link between the integral formulas from [53]and the combinatorial formulas from [44]97Canonical arc index of cable links(Hwa Jeong Lee and Hideo Takioka)

An open-book decomposition ofis a decomposition which has open half planes
as pages and the standard -axis as the binding axis. An arc presentation of a link
is an embedding of in finitely many pages of an open-book decomposition sothat each of these pages meets in a single simple arc. It is known that every, of a linkislink has an arc presentation [10]. The arc index, denoted by
the minimum number of pages needed for to be presented as an arc presentation.
It is known that the arc index equals the crossing number plus two for non-split
alternating links [1, 45, 56]. For a non-alternating prime link , we have the following$\mathbb{R}^{3}$

$z$

$L$

$L$

$L$

$\alpha(L)$

$L$

$L$

inequality [27, 45]:
$spread_{a}(F_{L}(a, z))+2\leq\alpha(L)\leq c(L)$ ,

9

$L$

139
where $F_{L}(a, z)$ is the Kauﬀman polynomial of and $c(L)$ is the crossing number of. It is important to determine the arc indices of non-alternating links. Since thearc indices of torus knots are determined in [16], we focus on cable links.
A grid diagram of a link is a link diagram which consists of vertical and horizontal
line segments with the properties that at each crossing the vertical line segment
crosses over the horizontal line segment and no two horizontal line segments arecollinear and no two vertical line segments are collinear. A grid diagram withvertical line segments is easily converted to an arc presentation on pages, and viceversa [10]. Therefore, the arc index of can be defined as the minimum number ofvertical line segments among all grid diagrams of
$L$

$L$

$n$

$n$

$L$

$L.$

$arrow$

Figure 5:$\alpha(G_{1}^{(3,16)})=\alpha(G_{2}^{(3,16)})=15,$ $\alpha(G_{3}^{(3,16)})=18$

for grid diagrams$G_{1},$ $G_{2},$ $G_{3}$of the righthanded trefoil

be a grid diagram of a knot $K$ andLetintegers with $p>$ O. In [35],we constructed an algorithm called the canonical $(p, q)$ -cabling algorithm of inorder to obtain a sharper upper bound of the arc index of the $(p, q)$ -cable link
of $K$ . Briefly, the procedure is given in Figure 5. We call the resulting diagramthe canonical grid diagram of. Let
obtained from and denote it bydenote the number of vertical line segments of. The canonical arc
$G$

$p,$

$q$

$G$

$K^{(p,q)}$$K^{(p,q)}$$G^{(p,q)}$$G$$G^{(p_{)}q)}$$\alpha(G^{(p,q)})$

10

140
index of$K^{(p,q)}$, denoted by$\alpha_{c}(K^{(p,q)})$$\alpha_{c}(K^{(p,q)})=\min${, is defined as follows:$\alpha(G^{(p,q)})|G$

is a grid diagram of $K$ }.

From the definition of the arc index and the canonical arc index, we have thefollowing inequality:$\alpha(K^{(p,q)})\leq\alpha_{c}(K^{(p_{)}q)})$.

Then we have a natural question:
Question 7.1 (H. J. Lee, H. Takioka). $\alpha(K^{(p,q)})=\alpha_{c}(K^{(p,q)})$ ?
It is shown in [35] that the answer of Question 7.1 is “yes” for infinite families of(2, q)-cable links of all prime knots with up to 8 crossings. Moreover, we see thata minimal grid diagram of $K$ leads to the equality above in this special case. The
most interesting view is whether any minimal grid diagram of $K$ gives the exact
value of arc indices of cable links of $K$ . So, we have natural questions:
Question 7.2 (H. J. Lee, H. Takioka). For two minimal grid diagrams $G,$ $G’$ of aknot $K$ , we have?
$\alpha(G^{(p,q)})=\alpha(G^{\prime(p,q)})$Question 7.3 (H. J. Lee, H. Takioka). If$K$ , then we have$G$

is a minimal grid diagram of a knot$\alpha_{c}(K^{(p,q)})=\alpha(G^{(p,q)})$8?

The state numbers for virtual knots(T. Nakamura, Y. Nakanishi, S. Satoh, Y. Tomiyama)
with ordinary crossings, which
A virtual knot diagram is a knot diagram in, and virtual crossings $(X)\cdot$ A virtual
are called real crossings
knot $K$ is an equivalence class of virtual knot diagrams under “generalized Reidemeister moves”’ (cf. [30]).
Let $D$ be a virtual knot diagram. A state of $D$ is a union of circles possiblywith virtual crossings obtained from $D$ by splicing all real crossings. A state issaid to be an -state if consists of circles. We denote by $s_{n}(D)$ the number of-states of D. The -state number for a virtual knot $K$ , denoted by $s_{n}(K)$ , is theminimal number of $s_{n}(D)$ for all possible virtual knot diagrams for $K$ (cf. [47]).
In [47], the following upper and lower bounds of $s_{n}(D)(n=1,2,3)$ are given:
For any virtual knot diagram $D$ with real crossings, it holds that $1\leq s_{1}(D)\leq$$\mathbb{R}^{2}$$D$$(\nearrow^{\backslash }\backslash , \backslash _{/}\nearrow)$

$S$

$S$

$S$

$n$

$n$

$n$

$n$

$D$

$r$

, and $0\leq s_{3}(D)\leq 3\cdot 2^{r-3}$ . Moreover, it is showedthat $r\leq s_{2}(D)$ if $r\geq 3$ . Virtual knot diagrams realizing the lower bound of each$s_{n}(D)(n=1,2,3)$ are characterized in [47] and [48].
$\frac{2^{r+1}-(-1)^{r+1}}{3},$$0\leq s_{2}(D)\leq 2^{r-1}$

Problem 8.1 (T. Nakamura, Y. Nakanishi, S. Satoh, Y. Tomiyama). Characterize
the virtual knot diagrams realizing the upper bound of each $s_{n}(D)(n=1,2,3)$ .
11

141
Namely, characterize the virtual knot diagrams $D$ witheach of$s_{2}(D)=2^{r-1}$ ,$s_{1}(D)= \frac{2^{r+1}-(-1)^{r+1}}{3},$

and

$r$

real crossings atisfying$\mathcal{S}$$s_{3}(D)=3\cdot 2^{r-3}.$

Examples of such virtual knot diagrams are given in [47] in terms of their Gauss
diagrams.
Let $D$ be a virtual knot diagram. We regard $D$ as the image of an immersion of acircleintowith crossing information at each double point. A Gauss diagram of$D$
is an oriented circle regarded as the preimage of the immersed circle with chords,
each of which connects the preimages of each double point corresponding to a real
crossing. A chord is oriented from the preimage of the over-crossing-point to that of
the under-crossing-point in the circle, and labeled by the sign of the corresponding
real crossing. Two chords of a Gauss diagramof $D$ is linked if their end-points
appear along the circle on alternately. A chord is free if it is not linked with anyother chords.
Letandbe the Gauss diagrams with chords as in Figure 6 (1) and (2),respectively. Then we see that
produces a virtual knot diagram $D$ satisfying$S^{1}$$\mathbb{R}^{2}$

$G$

$G$

$F_{r}$$F_{r}’$$r$$F_{r}$

81 $(D)= \frac{2^{r+1}-(-1)^{r+1}}{3},$$F_{r}’$

and$F_{r-1}$

with one free chord produce virtual knot

diagrams $D$ satisfying $s_{2}(D)=2^{r-1}$ . Moreover,with two free chords andwith three free chords produce virtual knot diagrams $D$ satisfying$F_{r-2}’$$F_{r-3}$$\mathcal{S}_{3}(D)=3\cdot 2^{r-3}.$

(1)

(2)

$F_{r}$$F_{r}’$

Figure 6

It is known that $s_{1}(K)\geq|V_{K}(-1)|=\det K$ for a classical knot $K$ , where $V_{K}(t)$ is
the Jones polynomial for $K$ and $\det K$ is the determinant of $K$ . As a generalizationof this fact, lower bounds of $s_{1}(K)$ for a virtual knot $K$ are given in terms of theJones polynomial and the Miyazawa polynomial (cf. [26, 40]) in [47].

Problem 8.2 (T. Nakamura, Y. Nakanishi, S. Satoh, Y. Tomiyama). Find a lower
bound for $s_{n}(K)(n\geq 2)$ by algebraic invariants, such as the Jones polynomial the
Miyazawa polynomial and so on.
J. Green [20] made a table of virtual knots with four real crossings or less. We
propose a fundamental problem.

Problem 8.3 (T. Nakamura, Y. Nakanishi, S. Satoh, Y. Tomiyama). Determine
for the virtual knots in Green’s table.

$s_{n}(K)(n\geq 1)$

Y. Tomiyama [57] determined $s_{1}(K)$ of several virtual knots in the table. In [47],and $s_{2}(K)$ of the virtual knots with three real crossings or less are determined.

$s_{1}(K)$

12

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9

Local and global properties of graphs(Takahiro Matsushita)

A graph is a pair $G=(V(G), E(G))$ where $V(G)$ is a set and $E(G)\subset V\cross V$ suchthat $(x, y)\in E(G)$ implies $(y, x)\in E(G)$ . A graph homomorphism from to $H$ isa map : $V(G)arrow V(H)$ such that $(f\cross f)(E(G))\subset E(H)$ . The existence problem
of graph homomorphisms between two graphs is a classical problem of graph theory,and this is closely related to the existence problem of -equivariant maps between
given two -spaces. For backgrounds of these topics, see [31].
In [38], -fundamental groups
and -covering maps are defined, which canbe applied to the existence problem of graph homomorphisms. There is a close
relation between -fundamental groups and -covering maps as is the case of thecovering space theory in topology.
To state the problems suggested here, we give the definition of -covering maps.
For $v\in V(G)$ , we write $N(v)$ for $\{w\in V(G)|(v, w)\in E(G)\}$ , and $N_{s}(v)$ for $s\geq 1$
is defined by $N_{1}(v)=N(v)$ , $N_{s+1}(v)= \bigcup_{w\in N_{s}(v)}N(w)$ . A graph homomorphism: $Garrow H$ is called an -covering map $(r\geq 1)$ if for any $v\in V(G)$ and with
$1\leq i\leq r,$ $p|_{N_{i}(v)}:N_{i}(v)arrow N_{i}(p(v))$ is bijective.
Let us observe some phenomena. Let be a positive integer, and a connected
graph such that $\# N(v)=n$ for any $v\in V(G)$ . Consider the following conditions.
$G$

$f$

$\mathbb{Z}_{2}$$\mathbb{Z}_{2}$$\pi_{1}^{r}(G)$

$r$

$r$

$r$

$r$

$r$

$i$

$r$

$p$

$G$

$n$

.

(a) $\#(N(v)\cap N(w))>n/2$ for any$v,$(b) $\#(N(v)\cap N(w))=1$ for any$w\in V(G)$ with $v\neq w$ and $w\in N_{2}(v)$$v,$$w\in V(G)$ with $w\in N_{2}(v)$.

If satisfies (a), then the diameter of is smaller than 4 (and hence, such a isfinite). If satisfies (b), thenmust be free.
From the above phenomena, we can observe that there might be close relations
between the (local” and the “global properties of graphs. Before suggesting questions, let us make the meaning of the (local” property of graphs clear. For a positive
integer , let us call the property (P) of graphs is -local if for a surjective -covering
map $p:Garrow H,$ satisfies (P) if and only if $H$ satisfies (P). The above properties(a) and (b) are 2-local properties in this definition.
$G$

$G$

$G$

$G$

$\pi_{1}^{2}(G)_{ev}$

$r$

$r$

$r$

$G$

Question 9.1 (T. Matsushita).
(1) Find -local properties such that a connected graph satisfying such a property
is finite.
$r$(2) Find -local properties such that if a connected graph $G$ satisfies such a property,
thenis hyperbolic.
$r$$\pi_{1}^{r}(G)$As one example of the above question (1), I suggest the following.
Question 9.2 (T. Matsushita). Is a connected graph2-local property finit$e^{}?$$\bullet$$\sup\{\# N(v)|v\in V(G)\}<\infty.$

13

$G$

satisfying both the following

143$\bullet$

For vertices$v,$$w\in V(G)$ such that $v\neq w$ ,we have $N(v)\neq N(w)$ and $\#(N(v)\cap$$N(w))\neq 1.$

These questions are related to the existence problem of graph homomorphisms.
Indeed, if a 2-local property (P) implies the finiteness for connected graphs, then theuniversa12-covering of the graph also satisfies (P), and hence is finite. This implies
that if a connected graph satisfies (P), thenis finite, and we have that thechromatic number of is not equal to 3 by the result of [38].
$G$$\pi_{1}^{2}(G)$

$G$

10

Essential tribranched surfaces in 3-manifolds(Takashi Hara)

Throughout this section let $M$ be a 3-manifold which is compact, connected,
irreducible and orientable. A closed subspace of $M$ is called a tribranchedsurface
if the following conditions are fulfilled:$\Sigma$(TBSO) the pairis locally homeomorphic to
is a closed subspace of defined as$(M, \Sigma)$$Y$$(\mathbb{C}\cross[0, \infty$), $Y\cross[O,$where$\infty$$\mathbb{C}$$Y=\{$$re^{\sqrt{-1}\theta}$$r\in[0, \infty)$,$\theta=0,$$\pm\frac{2}{3}\pi\}$;(TBSI) the intersection of and a suﬃciently small tubular neighbourhood ofin $M$ is homeomorphic to;(TBS2) each connected component ofis orientable.
$\Sigma$$C(\Sigma)$$Y\cross C(\Sigma)$$S(\Sigma)$

Here we denote by$C(\Sigma)$the closed subset of$\Sigma$(called the branch set of ) which cor$\Sigma$

responds to the subset $\{0\}\cross[0, \infty$ ) of
) under the identification in (TBSO),bythe complement of a suﬃciently small tubular neighbourhood ofin$\mathbb{C}\cross[0, \infty$$S(\Sigma)$$C(\Sigma)$, and bythe complement of a suﬃciently small tubular neighbourhood ofin $M.$
Now let us focus on a certain class of tribranched surfaces contained in 3-manifolds.
A tribranched surface in $M$ is said to be essential if it has following properties:(ETBSI) for each connected component $N$ of, the natural functorial homomorphism $\pi_{1}(N)arrow\pi_{1}(M)$ is not surjective;(ETBS2) for connected components $C,$
and $N$ of,
andrespectively, the natural functorial homomorphisms $\pi_{1}(C)arrow\pi_{1}(S)$ and$\pi_{1}(S)arrow\pi_{1}(N)$ are injective (ifthey exist);(ETBS3) there does not exist a connected component of which is contained in aball in $M$ or a collar of$\Sigma$$M(\Sigma)$$\Sigma$$\Sigma$$M(\Sigma)$$S$$C(\Sigma)$$S(\Sigma)$$M(\Sigma)$$\Sigma$$\partial M.$

The notion of essential tribranched surfaces is a natural generalisation of that ofessential
in a usual sense; indeed an essential surface is regarded as an essential tribranched surface with the empty branch set. Based upon geometry of character varieties of higher-dimensional representations and the theory of Bruhat-Tits$surface\mathcal{S}$

14

144
buildings, we may systematically construct essential tribranched surfaces containedin a 3-manifold $M$ , which may be regarded as a natural extension of the methodof Marc Culler and Peter B. Shalen in [11] (the main theorem of [23]; refer also to[22]). The first problem is whether or not our method provides a more sophisticatedway even in the construction of essential surfaces (without branch sets). Namely,

Question 10.1 (T. Hara, T. Kitayama). It is known that there exist essential
faces which one could not obtain utilising the method of Culler and[23]Shalen. Can one
construct such essential surfaces by applying the construction in
One great merit of our method is that we may apply this construction even-character varieties are of dimension zero; in
to 3-manifolds whose associatedparticular, we may apply it to (a certain class of) non-Haken manifolds! It is thus
in the nature of things that essential tribranched surfaces are expected to containfruitful information concerning topological properties of non-Haken manifolds.
$\mathcal{S}ur-$$\varphi$$SL_{2}$

Problem 10.2 (T. Hara, T. Kitayama). Extract topological information of (nonHaken) 3-manifolds from the information of essential tribranched surfaces containedin them.
Let us deal with a more concrete question concerning Problem 10.2. In the
low-dimensional topology, it is widely known that the procedures for cutting along
essential surfaces endow a Haken manifold with a structure called the Haken hierarchy. This simple observation leads us to the following naive question.

Question 10.3 (T. Hara, T. Kitayama). Do the procedures for cutting along essential tribranched surfaces2 endow a manifold (containing essential tribranchedsurfaces) with a certain structure like the Haken hierarch$y^{\prime p}$

The notion of essential tribranched surface itself is, however, quite new and rather
mysterious at the present, and therefore we might have to study topological properties of essential tribranched surfaces more deeply before trying Problem 10.2 or
Question 10.3.
Next we point out that essential tribranched surfaces are deeply related to thetheory of complexes of groups, as essential surfaces (in a usual sense) are to Bassand Serre’s theory of graphs of groups. Indeed we may associate a 2-complex of

in a canonical manner to an essential tribranched surface contained
in $M$ (see [23] for details). Contrary to graphs of groups, a 2-complex of groups
does not always come from a group action on (contractible) 2-complex (see [6,comes from an action of $\pi_{1}(M)$
Chapter III. ] and [9]). To guarantee thaton a contractible 2-complex, we should impose the following additional condition ongroups$\Sigma$$c(y_{\Sigma})$$a$$c(y_{\Sigma})$$C$$\Sigma$:(ETBS4) for each connected component $N$ ofmorphism $\pi_{1}(N)arrow\pi_{1}(M)$ is injective.

$M(\Sigma)$, the natural functorial homo-

we remark that, after cutting along an essential tribranched surface, the resulted manifold is equipped$\overline{2Here}$as a sutured manifold.
with a structure

15

145
We call an essential tribranched surface satisfying the additional condition (ETBS4)
a strongly essential tribranched surface. The additional condition (ETBS4) is arather algebraic (and ad hoc) condition, and thus we are interested in whether or
not there exists a topological criterion which distinguishes the notion of strong essentiality from that of essentiality.
$\Sigma$

Question 10.4 (T. Hara, T. Kitayama). Does there exist a topological (or geometrical)
characterisation of strongly essential tribranched surface$s^{}?$We remark that there is a geometric suﬃcient condition for a 2-complex of groups
to come from a group action on (contractible) 2-complex, which is called the nonpositive curvature condition (for details see [6, 9 The $(answer^{)}$ ’ to Question 10.4
might be related to such a kind of conditions.
We would like to end this section with a practical problem. Note that we may
not apply the method of [23] if the associated character variety is of dimension zero.
It is therefore crucial to know when its dimension is positive.
$a$Problem 10.5 (T. Hara, T. Kitayama). For a natural number greater than orequal to three, find a suficient condition for the-character variety to be ofdimension greater than or equal to one (as practical as possible).
Here we limit ourselves to topics around topological properties of essential tribranched surfaces for want of space. We shall deal with problems and questions
concerning actions of 3-manifold groups from the arithmetic viewpoint in [22].
$n$$SL_{n}$11

Invariants of homology 3-spheres motivated by the ChernSimons perturbation theory(Tatsuro Shimizu)

We denote by $Z^{KKT}$ the invariant of homology 3-spheres defined by G. Kuperberg and D. Thurston in [32] motivated by the Chern-Simons perturbation theory.
T. Watanabe’s invariant $Z^{FW}[62]$ and the invariant of [54] give alternative constructions of $Z^{KKT}$ . Let $Z^{LMO}$ be the LMO invariant [34].
Question 11.1. $Z^{KKT}=Z^{LMO}$ for rational homology 3 spheres 9$\tilde{Z}$A remarkable progress toward this question is given by D. Moussard. She provedin [46] that $Z^{LMO}$ and $Z^{KKT}$ have same ability to distinguish two rational homology 3-spheres in the sense that, for rational homology 3-spheres $M$ and $N$ with$|H_{1}(M;\mathbb{Z})|=|H_{1}(N;\mathbb{Z})|$ and for any$n\in \mathbb{N},$$(Z_{k}^{LMO}(M)=Z_{k}^{LMO}(N)$for all $k\leq n)\Leftrightarrow(Z_{k}^{KKT}(M)=Z_{k}^{KKT}(N)$ for all $k\leq n)$ .

Let be a rational homology 3-sphere. The topological invariant $Z_{n}^{KKT}(Y)$ is asum of the principal term depending on a framing of and the correction term tocancel out the ambiguity of the choice of . The correction term is given by,
where
is the signature defect of andis a constant independent of and$Y$$\tau$$\tau$$\sigma_{Y}(\tau)$$Y$$\delta_{n}\sigma_{Y}(\tau)$$\delta_{n}$$\tau$$\tau$

16

146; see [36] for the definition of . Kuperberg and Thurston conjectured in [32] that
this correction term is vanishing for any $n>1.$$Y$$\delta_{n}$Conjecture 11.2 ([32, Conjecture 6$\delta_{n}=0$

for any $n>1.$

The next question may be related to the singularity theory of smooth maps. Let
be a closed compact oriented 4-manifold with a metric such that $\chi(X)=0$ , where$\chi(X)$ is the Euler characteristic of $X$ . Take a unit vector field
on $X$ and denote by$T^{v}X$$TX$. Let
be “generic vector fields ofandthe normal bundle of in$T^{v}X$$\{x\in
X|\dim\langle\beta_{1}(x),\beta_{2}(x),\beta_{3}(x)\rangle=1\}\subset X$ has,
set. For genericthe,,a structure of a compact oriented -dimensional manifold. Here.
,and
So we can countspanned byis the vector subspace of. The
the number of point of this set with sign. We denote it by$Z^{FW}$
implies (see Appendix of [54]) that
construction of the correction term of or. It follows from this fact andfor any generic$\chi(X)=0$ thatis divisible by 6.
$X$$\gamma$$\beta_{1},$$\beta_{3}$$\beta_{2}$$\gamma$$\beta_{1},$$\beta_{2},$$\beta_{3}$$0$$\langle\beta_{1}(x)$$(T^{v}X)_{x}$$\beta_{1}(x)$$\beta_{2}(x)$$\beta_{2}(x)$$\beta_{3}(x)\rangle$$\beta_{3}(x)$$\langle\langle\beta_{1},$$\beta_{2},$$\beta_{3}\rangle\rangle\in \mathbb{Z}$$\tilde{Z}$$\langle\langle\beta_{1},$$\beta_{2},$$\beta_{3}\rangle\rangle=3Sign(X)$$\langle\langle\beta_{1},$$\beta_{2},$$\beta_{1},$$\beta_{2},$$\beta_{3}$$\beta_{3}\rangle\rangle$

Question 11.3 (T. Shimizu). What is a topological interpretation of a reason why$\langle\langle\beta_{1},$$\beta_{2},$$\beta_{3}\rangle\rangle\in 6\mathbb{Z}^{\varphi}$

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