115

数理解析研究所講究録
第2004巻 2016年 115-129

Problems on Low‐dimensional Topology, 2016
Edited by T. Ohtsukil

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 18‐20, 2016.
$\zeta$Contents
1

Heegaard Floer homology for embedded bipartite graphs22

Heegaard Floer homology of knots and 3‐manifolds

33

Modifying constructions of Lagrangian and Heegaard Floer theory

5

4

Braid groups and \mathrm{C}^{p} ‐groups

6

5Complex of surfaces of a 4‐manifold and the adjunction inequalities

7

6

Surface‐links and marked graph diagrams

8

7

Surface‐links which bound immersed handlebodies

118Morse‐Novikov numbers of surface‐links

12

1Research Institute for Mathematical Sciences, Kyoto University, Sakyo‐ku, Kyoto, 606‐8502, JAPAN
Email:

tomotada@kurims.kyoto‐u.ac.jp

The editor is partially supported by JSPS KAKENHI Grant Numbers 16\mathrm{H}02145 and 16\mathrm{K}13754.

1

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1

Heegaard Floer homology for embedded bipartite graphs(Yuanyuan Bao)
For a link L in S^{3} its link diagram in S^{2} with a basepoint determines a Hee‐gaard diagram. Ozsváth and Szabó showed in [34] that (1) the generators of the
Heegaard Floer chain complex are given by the Kauffman states of the diagram;(2) the Maslov grading and Alexander grading can be calculated in an explicit andcombinatorial way; and as a corollary (3) the Heegaard Floer homology (hat ver‐sion) is explicitly determined by the signature and the Alexander polynomial of thegiven link. However in general, a combinatorial description of the differential of thecomplex is unknown.
,\swarrow\leftrightar ow\mapsto\mathrm{t}\}'\mathrm{p}\mathrm{e}1\mathrm{t}_{\mathrm{V}\mathrm{t}\ }\mathrm{e} II

For a balanced bipartite graph G_{V_{1},V_{2}} in S^{3} we similarly provide a Heegaard
diagram for it from its diagram in S^{2} with a basepoint. We also prove that (1) thegenerators of the Heegaard Floer chain complex for the graph are given by the statesof the diagram (Figure above); (2) the Alexander polynomial of the graph can be
expressed as a state sum. We may ask the following questions, which we suggest to,

answerin order.

Question 1.1 (Y. Bao). Can any two states be connected by transpositions of type I
and II? Is it possible to calculate the (relative) Maslov grading and Alexander grading
combinatorially9 For alternating (there is no standard definition) bipartite graphs,
is the Heegaard Floer complex completely determined by the Alexander polynomial(up to overall shifts of the gradings) /?
In a recent paper [36], from a knot diagram, Ozsváth and Szabó constructed abigraded chain complex over \mathbb{F}[U] the homology of which is shown to be isomorphic
to the knot Floer homology (minus version). It is freely generated by the Kauffman
states, and its differential is defined algebraically, built on bordered Floer homology.
We may ask the following question.
,

Question 1.2 (Y. Bao). For a bipartite graph, or a general graph, is it possible toconstruct such a complex2

2

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Heegaard Floer homology of knots and 3‐manifolds

2(Motoo Tange)
The hat version of Heegaard Floer homology is algorithmically computable using
the Heegaard diagram. This result is due to [37]. Another development of this topic

is the result [33] by Ozsváth, Stipsicz and Szabó. Also, all types of Heegaard Floer
homology with coefficients \mathbb{Z}/2\mathbb{Z} is computable combinatorially [27].

(M. Tange). Find an algorithm to compute HF^{+}(Y, \mathfrak{s}) HF^{-}(Y, \mathfrak{s})Problem 2.1
with \mathb {Z},coeff cients.

The Ozsváth‐Szabós d‐invariant is difficult to compute for general 3‐manifold.
Indeed, for suitable graph manifolds (Nemethis algorithm [29]) or S^{1} ‐bundle over a

surface.

Problem 2.2(M. Tange). Find d ‐invariant formula for hyperbolic 3‐manifold.

Let K be a knot with annulus presentation.

For the definition of annulus pre‐sentation, see [1]. Let A^{n}(K) be the n ‐fold annulus twist of K along the embedding
annulus. The annulus twist is useful method to construct a pair of the diffeomorphic
0 ‐surgery or diffeomorphic 0 ‐framed 2‐handlebody.
If two knots K and K' are concordant, then 0 ‐surgeries of K and K' are homology
cobordant. In general, we do not know whether K and A^{n}(K) are concordant or[12] gave a formula of CFK of Whitehead double D_{+}(K, n) from thatAs an analogy, the following question is considered:not. Hedden

of K

.

Problem 2.3(M. Tange). Compute CFK^{\infty}(A^{n}(K)) from CFK^{\infty}(K)In the case of K=6_{3} , the annulus twist of 63 has thesame$\tau$.

‐invariant,

i. e.,$\tau$(6_{3})= $\tau$(A^{n}(6_{3})) holds for any n.
Question 2.4 (M. Tange). For any n does the equality $\tau$(K)= $\tau$(A^{n}(K)) hold, ingeneral!?
Question 2.5 (M. Tange). Are the d ‐invariants of the 0 ‐surgeries of K and K'
equal'? Or are the d ‐invariants of the branched covers of K and K' equal9
For a knot K in S^{3} and an integer n , let S_{n}^{3}(K) denote the 3‐manifold obtained

from S^{3} by n ‐surgery along K.

A. Levine in [40] asks the following question:

If K_{1} is concordant to K_{2_{J}} then for all n, S_{n}^{3}(K_{1}) is homology cobordanttoS_{n}^{3}(K).

Is the converse tru e^{\ell}

Here we put some stronger problem:

Problem 2.6
invariant for(M. Tange). Find non‐concordant knots K_{1} and K_{2} with same d‐

S_{n}^{3}(K_{1}) and S_{n}^{3}(K_{2}) for any integer n.

Due to [35] knot Floer homology detects the Seifert genus:

g(K)=\displaystyle \max\{s|\overline{HFK}(S^{3}, K, s)\neq 0\}.
3

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The $\tau$ ‐invariant gives the bound for 4‐ball genus as follows:| $\tau$(K)|\leq g_{4}(K).

The following question seems fundamental, but the answer is not known.

Question 2.7 (M. Tange). Does knot Floer homology detect the 4‐ball genus 9
The knot Floer homology also detects fiberedness of a knot due to [30].
K is a fibered knot

\Leftrightarrow\overline{HFK}(K, g(K))\cong \mathbb{Z}.

What is a geometric property characterizing the knot Floer homology
for i<g(K) ?

\overline{HFK}(K, i)

Suppose that K is a lens space (or \mathrm{L} ‐space) knot (^{\mathrm{d} \Leftrightar ow^{\mathrm{e}\mathrm{f} \exist p\in \mathb {Z} such that S_{p}^{3}(K) is
lens space (or \mathrm{L}a‐space)). Then any \overline{HFK}(K, i) is isomorphic to \mathb {Z}or\{0\}.

Question 2.8 (M. Tange). What is a geometric property characterizing the isomor‐

phism \overline{HFK}(K, i)\cong \mathbb{Z}'?
Manolescu in[26] defined Pin(2)‐equivariant Seiberg‐Witten Floer homology.

Here we raised the following problem.

Problem 2.9 (M. Tange). Define Pin(2)‐equivariant Heegaard Floer homology which
is isomorphic to Manolescus Pin(2)‐equivariant Seiberg‐Witten Floer homology.

Let f : X\rightarrow S^{1} be a circle valued Morse function. Let $\tau$_{\mathrm{t}\mathrm{o}\mathrm{p} denote the torsionof cell complex of the infinite cyclic cover \tilde{X} with respect to f Let $\tau$Morse denote.

the torsion of Morse complex of the infinite cyclic cover \tilde{X} with respect to f

.

Morse

complex is generated by the critical points of the Morse function and the differentials
aredefined by counting of trajectories between critical points. Then Hutchings andLee proved the following formula:$\tau$_{\mathrm{M}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{e} (t)\cdot $\zeta$(t)=$\tau$_{\mathrm{t}\mathrm{o}\mathrm{p} .
The difference$\zeta$(t) between two Reidemeister torsions is a zeta function of the

dynamical system on a level set. The right‐hand side is equivalent to a summationof Seiberg‐Witten invariants on X due to the result [28] by Mark.
As a Hutchings‐Lee type formula, Goda, Matsuda, and Pajitnov in [9] proved theformula below. Let K be a knot in S^{3} and f : S^{3}-K\rightarrow S^{1} a circle valued Morsefunction. Let v be a half transversal flow for f and $\tau$_{v} denote the Novikov torsion forv
Let R be the regular surface of f and h : R\rightarrow R a monodromy map generalized
to even any non‐fiUered knot. Let and $\zeta$_{h}(t) the zeta function of dynamical system
h Then the following equality holds:.

.

$\tau$_{v}(t)$\zeta$_{h}(t)=\displaystyle \frac{\triangle_{K}(t)}{t-1}.
Problem 2.10

(M. Tange). Interpret this formula by using some Floer theory.
4

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Heegaard Floer counterpart of the right‐hand side is the Heegaard Floer homology
HFK^{-}(K) This problem implies a decomposition of HFK^{-}(K) into two somehomology theories for Novikov torsion and zeta function. The $\zeta$ ‐part would be
symplectic Floer homology for mapping classes.
Casson invariant is the first term of LMO invariant Z^{LMO}(Y) of a homology
3‐sphere Y To the higher terms of this invariant we have not found geometricmeanings. Here we propose the following problem.
.

.

Problem 2.11some

3

(M. Tange). By deforming Heegaard (or instanton) Floer theory insense, find the higher terms in it again.

Modifying constructions of Lagrangian and Heegaard Floer
theory(Kaoru Ono)
For a pair of Lagrangian submanifolds L_{1}, L_{2} in a closed symplectic manifold,
Lagrangian Floer homology HF_{*}(L_{1}, L_{2}) is defined from a chain complex generated
by intersection points of L_{1} and L_{2} whose differential counts pseudo‐holomorphic
disks; for details see [6, 7]. Further, in [7], it is extended to HF_{*}((L_{1}, b_{1}), (L_{2}, b_{2}))for bounding cochains b_{i} of L_{i}(i=1,2) For a 3‐manifold M the instanton Floer
homology of M is defined by (infinite dimensional) Morse theory for the Chern‐
.

,

((

Simons functional on the space of SU(2) connections on M ; see e.g.

[5].

Motivatedby the instanton Floer homology, for a 3‐manifold M Heegaard Floer homology of
M is defined as Lagrangian Floer homology associated to a Heegaard diagram of,

M ; for details see e.g.

[3].

Question 3.1 (K. Ono). There are some constructions in Lagrangian Floer theory.
Is it possible to consider such constructions in Heegaard Floer theory and apply themto low dimensional topology ?
lConstruction 1(bounding cochain).

Foratopological space, (co)homology the‐\mathrm{o}\mathrm{r}\mathrm{y} can be twisted by a local system. In particular, we have Morse (co)homology
with coefficients in a local system. Under certain conditions, we can also construct

Lagrangian Floer complex twisted by local systems on Lagrangian submanifolds.
This construction is a part of the following story. Let (L_{1}, L_{2}) be \mathrm{a} (transversal)
pair of Lagrangian submanifolds in a closed (or (tame) symplectic manifold (X, $\omega$)
In general, Floer chain complex for (L_{1}, L_{2}) is not defined. The obstruction isformulated in terms of filtered A_{\infty} ‐algebras associated with L_{i}, i=1 2. If Maurer‐
Cartan equations in these filtered A_{\infty} ‐algebras have solutions (we call them bounding
cochains or Maurer‐Cartan elements), we can modify the definition of the boundary
operator to obtain a chain complex. (More generally, we can work with a pair of
weak bounding cochains (weak Maurer‐Cartan elements) with the same potential.

,

value.)

We can introduce an equivalence relation among(weak) bounding cochains sothat the cohomology groups associated to equivalent (weak) bounding cochains are5

120

isomorphic. The notion of augmentation in the setting of contact homology is ananalog of bounding cochains.
There is also a construction, called bulk deformations [8]. In fact, a special kind of
bulk deformations had already used in Heegaard Floer theory from the early stage.
Construction 2(filtration by the action functional).

Floer complex is a kind of

Morse‐Novikov complex associated to so‐called action functional (corresponding to

the Chern‐Simons functional for the instanton Floer

homology), which naturally
Using this filtration, one can obtain essen‐tial critical values of the action functional. (In the case of Morse theory, essential
critical values mean critical values corresponding to non‐zero homology classes.)
In symplectic Floer theories, it provides useful information such as spectral invari‐ants [32] for Hamiltonian diffeomorphisms, torsion exponents in Lagrangian Floer
induces a filtration on the complex.

(co)homology, etc.

Braid groups and \mathrm{C}^{p} ‐groups

4(Yuta Nozaki)
Let G be a group, and let p be a positive integer. As in [31], we define \mathrm{C}^{p}(G) tobe the subgroup of G generated by the set[g, h]=ghg^{-1}h^{-1}.

In other words,\{g^{p}|g\in G\}\cup\{[g, h]|g, h\in G\} where,\mathrm{C}^{p}(G) is the kernel of the natural projection

G\rightar ow G_{\mathrm{a}\mathrm{b} /pG_{\mathrm{a}\mathrm{b} where G_{\mathrm{a}\mathrm{b} denotes the abelianization of G.
,

When there is a p ‐fold cyclic covering (S^{3}, K)\rightarrow(L(p, q), K') for a knot K in

S^{3} and a knot K' in the lens space L(p, q) it is shown in [31] that the knot group,

G(K) is isomorphic to \mathrm{C}^{p}($\pi$_{1}(L(p, q)\backslash K

since the following sequence is exact,

G(K)\rightarrow$\pi$_{1}(L(p, q)\backslash K')\rightarrow \mathbb{Z}/p\mathbb{Z}.
Therefore, for an arbitrary knot K the following question naturally arises, which
discussed in [31]. Here, as in [31], we call a group G a \mathrm{C}^{p} ‐group if there exists
G' such that G is isomorphic to \mathrm{C}^{p}(G')
,

was

.

Question 4.1 (Y. Nozaki). Let K be a knot. Is G(K) a \mathrm{C}^{p}-group2
Remark. For a given knot K in S^{3} it is a non‐trivial problem to determine whether,

there is a knot K' in L(p, q) such that K is isotopic to the preimage of K' by the

projection S^{3}\rightarrow L(p, q) We note that, if such a K' exists, G(K) is a \mathrm{C}^{p} ‐group.
Hence, if we can show that G(K) is not a \mathrm{C}^{p} ‐group, it follows that such a K' does.

not exist.

Remark. Hartley [10] gave a list of possible free period of prime knots K with up to
10 crossings. Such a K can be obtained as the preimage of a knot in L(p, q).

Further, we consider the corresponding problem for braid groups.
Problem 4.2\mathrm{C}^{p} ‐group /?

(Y. Nozaki). Let p be odd and n\geq 3

6.

Is the nth braid group B_{n}

a

121

[31] that, if G is a \mathrm{C}^{p} ‐group and there is an epimorphism G\rightarrow

It is known in

G' whose kernel isa characteristic subgroup of G
then G' is also an \mathrm{C}^{p} ‐group.
Here, a characteristic subgroup of G is a subgroup which is invariant under all
automorphisms of G Let \mathfrak{S}_{n} be the nth symmetric group. The kernel of thenatural homomorphism B_{n}\rightarrow \mathfrak{S}_{n} is the pure braid group, which is a characteristic
subgroup of B_{n} Since \mathfrak{S}_{n} is not a \mathrm{C}^{p} ‐group for even p (as shown in [31, Example2.8]), it follows that B_{n} is not a \mathrm{C}^{p} ‐group for even p On the other hand, B3 is a
,

.

.

.

\mathrm{C}^{p} ‐group for p with \mathrm{g}\mathrm{c}\mathrm{d}(p, 6)=1.

Remark. Let X_{n} be the configuration space of n distinct points in \mathbb{R}^{2} The braid
group B_{n} is isomorphic to $\pi$_{1}(X_{n}/\mathfrak{S}_{n}) Problem 4.2 is related to a problem to find
.

.

an

5

appropriate space whose p‐fold cyclic cover is homeomorphic to X_{n}/\mathfrak{S}_{n}.

Complex of surfaces of a 4‐manifold and the adjunction
inequalities(Hokuto Konno)$\zeta$The notion of thecomplex of curves of a surface was introduced by Harvey[11] in the 1980\mathrm{s} and has been studied from the viewpoint of the Teichmüller space
and the action of the mapping class group. The complex of curves (also called curve
complex) of a surface S is defined to be the simplicial complex whose vertices are,

the isotopy classes of essential simple closed curves on S and whose simplices arespanned by collections of such curves which can be realized disjointly.
A4‐dimensional analog of this notion, namely,complex of surfaces was intro‐

duced by Mikio Furuta.2

Definition (M.

Furuta) Let X be an oriented, closed smooth 4‐manifold. The
complex of surfaces \mathcal{K}=\mathcal{K}(X) of X is the abstract simplicial complex defined asfollows:

The set of vertices V(\mathcal{K}) is given as the set of smooth embeddings of surfaces

with self‐intersection number zero:V(\mathcal{K}):=\{ $\Sigma$\mapsto X|[ $\Sigma$]^{2}=0\}.
Here we consider only oriented,
vertexclosed, connected surfaces.
$\Sigma$.
bybriefly( $\Sigma$\mapsto X)\in V(\mathcal{K})For k\geq 1 , a collection of (k+1) vertices $\Sigma$_{0}

if and only if $\Sigma$_{0} ,

.

.

.

,

,

.

.

.

,

We denote each$\Sigma$_{k}\in V(\mathcal{K}) spans a k ‐simplex$\Sigma$_{k} are disjoint.

In the above definition of thecomplex of surfaces, we do not consider the iso‐On the other hand, in the same way as
the definition of the complex of curves, one can define an abstract simplicial com‐plex whose vertices are the isotopy classes of embeddings of surfaces and whose
topy classes of embeddings of surfaces.

2The author heard this definition from Mikio Furuta in private communication in April 2015.

7

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simplices are spanned by collections of such isotopy classes which can be realized
disjointly. However, to give the following application to the adjunction inequalities
using Seiberg‐Witten theory, the first definition of the complex of surfaces might
be appropriate.
One can easily see that \mathcal{K}(X) is contractible for any X

Thus it is natural to.

seek a significant subcomplex of \mathcal{K}(X) having non‐trivial homotopy type.

Definition (H.

Konno) Let \mathfrak{s} be a spin \mathrm{c} structure on X Then, the complex of
surfaces violating the adjunction inequality \mathcal{K}_{V}=\mathcal{K}_{V}(X,\mathfrak{s}) is the subcomplex of\mathcal{K}(X) defined as the set of vertices is given by.

V(\displaystyle \mathcal{K}_{V}) :=\{ $\Sigma$\in V(\mathcal{K})|\max\{- $\chi$( $\Sigma$), 0\}<|c_{1}(\mathfrak{s})\cdot[ $\Sigma$]|\}and having the induced structure of an abstract simplicial complex from \mathcal{K}.

We showed that, for any k\geq 0 there exists infinitely many pairs (X, s) satisfying,\tilde{H}_{k}(\mathcal{K}_{V}(X,\mathfrak{s});\mathbb{Z})\neq 0.
This result gives a classical application; we can drive certain adjunction inequalities
for surfaces embedded to 4‐manifolds whose Seiberg‐Witten invariants vanish.
On the other hand, we do not know any example of (X, s) and k with

0 except for the case of V(\mathcal{K}_{V}(X,\mathfrak{s}))=\emptyset.

Problem 5.1 (H. Konno). Find an example of (X, s) and k with
0 and V(\mathcal{K}_{V}(X, \mathfrak{s}))\neq\emptyset.

\tilde{H}_{k}(\mathcal{K}_{V}(X, \mathfrak{s});\mathbb{Z})=\tilde{H}_{k}(\mathcal{K}_{V}(X, \mathfrak{s});\mathbb{Z})=The complex of curves is used to describe the end of the moduli space of complex
on the base surface.
On the other hand, the complex \mathcal{K}_{V} is used to(stretching of neighborhoods of embedded surfaces in 4‐manifold. The
Seiberg‐Witten equations on the stretched neighborhoods play a key role to consider
the adjunction inequalities. One natural limit of this stretching is a non‐compact
4‐manifold whose ends are given by the cylinders, i. e., product of the embedded
surfaces, S^{1} and the half line.

structuresdescribe,

Problem 5.2 (H. Konno). For a closed 4‐manifold, construct the moduli space ofacertain structure whose end is given by 4‐manifolds with cylindrical ends. Describe

the end of the moduli space in terms of the complex of surfaces.

Surface‐links and marked graph diagrams

6(Sang Youl Lee)A surface‐link is a closed 2‐manifold smoothly (or piecewise linearly and locallyflatly) embedded in \mathbb{R}^{4}.

Two surface‐links

ambient isotopic.

A marked graph diagram (or
some

are

said to beequivalent if they are

ch‐diagram) is a link diagram in \mathbb{R}^{2} possibly with

4‐valent vertices equipped with markers:

8\ovalbox{\t smal REJ CT}.

An oriented marked graph123

diagram is a marked graph diagram in which every edge has an orientation suchthat each marked vertex looks like

3 $\xi$or\geq\leq (seeFigure 1).

For a given (oriented) marked graph diagram D , let L_{-}(D) and L_{+}(D) be classical(oriented) link diagrams obtained from D by replacing each marked vertex

)(

and

,

Xwith

respectively (see Figure 1). An (oriented) marked graph diagram D is

said to \mathrm{b}\mathrm{e}admissible if both resolutions L_{-}(D) and L_{+}(D) are diagrams of (oriented)

trivial links.

Figure 1:

A marked graph diagram and its resolutions

S. J. Lomonaco, Jr. [25] and K. Yoshikawa [39] introduced a method of describing

surface‐links using marked graph diagrams.

Indeed, every surface‐link \mathcal{L} is repre‐
Moreover, if D is an admissi‐

sented by an admissible marked graph diagram D.

ble marked graph diagram representing a surface‐link \mathcal{L} , then one can construct asurface‐link \mathcal{L}_{D} from D in a canonical way such that \mathcal{L}_{D} is equivalent to \mathcal{L}.
$\Gamma$_{1} :\leftarrow^{\vec{}}$\Gamma$ í :\leftarrow^{\vec{}}$\Gamma$_{2} :\vec{\leftarrow}$\Gamma$_{3} :\leftarrow^{\rightarrow}$\Gamma$_{4} :\leftarrow^{\vec{}}$\Gamma$_{4}' :\backslah\ovalbox{\t smal REJ CT}_{-\nearow'\leftarrow^{\vec{}}\supset\supset\mathfrak{D}\mathrm{C}/*-\aleph$\Gamma$_{5} :Figure 2: Yoshikawa moves of type I
The Yoshikawa moves for oriented marked graph diagrams are the local moves

9

124

$\Gamma$_{6} :\leftarrow^{\rightarrow}$\Gamma$_{6}' :\vec{-}\supset\supset$\Gamma$_{7} :$\Gamma$_{8} :\leftarrow^{\rightarrow}

Figure 3:$\Gamma$_{1}

,

.

.

.

,

$\Gamma$_{5} (Type I) and $\Gamma$_{6}

,

.

.

.

,

Yoshikawa moves of type II$\Gamma$_{8} (Type II) illustrated in Figures 2 and 3. Let\mathfrak{S}=\{$\Gamma$_{1}, $\Gamma$\'{i})$\Gamma$_{2}, $\Gamma$_{3}, $\Gamma$_{4}, $\Gamma$_{4}', $\Gamma$_{5}, $\Gamma$_{6}, $\Gamma$_{6)}'$\Gamma$_{7}, $\Gamma$_{8}\}.
It is known that two admissible marked graph diagrams represent equivalent surface‐links if and only if theyaretransformed into each other by

11 Yoshikawa moves in \mathfrak{S}[22 ,afinite sequence of20, 38 ] Therefore any oriented surface‐link can be
represented by an oriented marked graph diagram [25, 39], and such a representation
diagram is unique up to the Yoshikawa moves in \mathfrak{S} For unoriented surface‐links.

.

( i.e.

,

non‐orientable surface‐links or orientable surface‐links without orientations),

the Yoshikawa moves in \mathfrak{S} forgetting the orientations are enough to describe theirmarked graph representations [19, 21, 22, 38].

On the other hand, it is proved that if $\Gamma$\in \mathfrak{S}-\{$\Gamma$_{5}, $\Gamma$_{8}\} then $\Gamma$ is independent,

from the other moves in \mathfrak{S}[21].

If the answers of the following two questions are allaffirmative, then \mathfrak{S} is a minimal generating set for oriented Yoshikawa moves.

Question 6.1 (J. Kim, Y. Joung, S. Y. Lee [21]). Is the Yoshikawa move $\Gamma$_{5} inde‐
pendent from the other moves in \mathfrak{S} ?
Question 6.2 (J. Kim, Y. Joung, S. Y. Lee [21]). Is the Yoshikawa move $\Gamma$_{8} inde‐
pendent from the other moves in \mathfrak{S}^{9}Let \mathcal{L} be a surface‐link and let D be a marked graph diagram of \mathcal{L} Let |V(D)| and|C(D)| denote the number of all marked vertices and classical crossings in D respec‐tively. In [39], Yoshikawa introduced the ch‐index, denoted by \mathrm{c}\mathrm{h}(\mathcal{L}) of a surface‐link\mathcal{L} which is defined to be the minimum number \displaystyle \mathrm{c}\mathrm{h}(\mathcal{L})=\min_{D\in \mathcal{D}}(|V(D)|+|C(D)|)
where \mathcal{D} denotes the set of all marked graph diagrams representing \mathcal{L} Clearly, \mathrm{c}\mathrm{h}(\mathcal{L})
.

,

,

,

,

.

is an ambient isotopy invariant of \mathcal{L}surface‐links with ch‐index \leq 10[39]Problem 6.3.

.

Using the terminology, he gave a table of 23

So it is natural to raise the following problem.

(S. Y. Lee). Create a complete table of admissible marked graph dia‐grams representing surface‐links with ch‐index \geq 11.
10

125

Up to now, many invariants for surface‐links have been defined by using various
representations of surface‐links, for example, broken surface diagrams, 2‐dimensional
braids, charts, etc. So the following problem can be considered.
Problem 6.4 (S. Y. Lee). How to compute known invariants for surface‐links using
marked graph diagrams./

In [25], S. J. Lomonaco, Jr. used marked graph diagrams to calculate the surface‐link groups. In [2], S. Ashihara gave a method of calculating the fundamentalbiquandles of surface links from their marked graph diagrams and Y. Joung, J.
Kim and S. Y. Lee compute the Alexander biquandles of oriented surface‐links via

marked graph diagrams in [20].

Recently, it is also shown that the quandle cocycle

invariants for surface‐links can be computed by using marked graph diagrams [17].

The answers of the following problems would enrich the theory of surface‐links.

Problem 6.5(S. Y. Lee). Construct new invariants for surface‐links with marked

graph diagrams.
Especially, the following problem is important.
Problem 6.6

marked graph(S. Y. Lee). Construct polynomial invariants for surface‐links withdiagrams which can be computed by recursive rules (skein relation)and categorifications.

So far, there have been several attempts to construct new invariants with marked
graph diagrams [13, 14, 15, 23, 24]. Finally, one may ask the following questions.

Question 6.7 (S. Y. Lee). Is it possible to construct quantum invariants for surface‐links with marked graph diagrams /?
Question 6.8 (S. Y. Lee). Is it possible to construct a surface‐link (co)homology
with marked graph diagrams.
Surface‐links which bound immersed handlebodies

7(Kengo Kawamura)An immersed surface‐link or simply a surface‐link means a closed oriented sur‐face generically immersed in \mathbb{R}^{4}surface‐link.

.

When it is embedded, we also call it an embedded

Two surface‐links areequivalent if there is an orientation‐preserving
diffeomorphism f : \mathbb{R}^{4}\rightar ow \mathbb{R}^{4} sending one to the other preserving their orientations.
A surface‐link is said to be ribbon if it is equivalent to a surface‐link which bounds
immersed handlebodies in \mathbb{R}^{4} whose multiple point set consists of ribbon singulari‐ties.

(Note that ribbon surface‐links are embedded surface‐links.) A surface‐link issaid to be ribbon‐clasp if it is equivalent to a surface‐link which bounds immersed

handlebodies in \mathbb{R}^{4} whose

multiple point set consists of ribbon singularities andclasp singularities; for details, see [16].

11

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A chord graph (O; $\alpha$) is a spatial trivalent graph which consists of a trivial link O

and disjoint simple arcs $\alpha$ spanning O. A chord diagram C(O; $\alpha$) is a diagram of achord graph (O; $\alpha$)

from

a

chord

It is known [18] that every ribbon surface‐link can be obtained.

graph (O; $\alpha$) up to equivalence.

The resulting ribbon surface‐link,denoted by F(O; $\alpha$) is obtained from a trivial 2‐link whose equator is O by 1‐handle,

surgeries along 1‐handles h( $\alpha$) whose cores are $\alpha$ A ribbon surface‐link F(O; $\alpha$)
equivalent to a ribbon surface‐link F(O';$\alpha$') if there is an equivalence
f : \mathbb{R}^{4}\rightar ow \mathbb{R}^{4} sending F(O; $\alpha$) to F(O';$\alpha$') and meridian curves of h( $\alpha$) to null‐homotopic curves in F(O';$\alpha$')\cup h($\alpha$') It is proved in [18] that two ribbon surface‐links F(O; $\alpha$) and F(O';$\alpha$') are faithfully equivalent if and only if the chord diagrams.

is faithfully.

C(O; $\alpha$) and C(O';$\alpha$') are related by a finite sequence of certain moves.
We generalize above arguments as follows. A chord graph (O\cup H; $\alpha$) is a spatial

trivalentgraph which consists of a split union of a trivial link O and Hopf links
simple arcs $\alpha$ spanning O\cup H. A chord diagram C(O\cup H; $\alpha$) isa diagram of a chord graph(O\cup H; $\alpha$) It can be seen that every ribbon‐clasp
surface‐link can be obtained from a chord graph (O\cup H; $\alpha$) up to equivalence. The
resulting ribbon‐clasp surface‐link, denoted by F(O\cup H; $\alpha$) is obtained from an
M ‐trivial 2‐link (for details; see [16]) whose equator is O\cup H by 1‐handle surgeries
along 1‐handles h( $\alpha$) whose cores are \mathrm{a} We similarly define a faithful equivalence
for ribbon‐clasp surface‐links. Then, we ask whether an analogous result holds.
H , and disjoint

.

,

.

Problem 7.1 (K. Kawamura). Find certain moves for chord diagrams C(O\cup H; $\alpha$)
which generate the faithful equivalence on ribbon‐clasp surface‐links F(O\cup H; $\alpha$).

This problem is a specialized version of the following problem.
Problem 7.2 (K. Kawamura). Find certain moves for chord diagrams C(O\cup H; $\alpha$)
which generate the equivalence on ribbon‐clasp surface‐links F(O\cup H; $\alpha$)8.

Morse‐Novikov numbers of surface‐links(Hisaaki Endo and Andrei Pajitnov)
A 2‐knot is a smoothly embedded 2‐sphere in S^{4} A Morse function f : S^{4}\backslash
K\rightarrow S^{1} on the complement to a 2‐knot K is called strongly minimal if its number.

of critical points m_{p}(f) of index p is minimal possible for every p.

The Morse‐

Novikov number \mathcal{M}\mathcal{N}(K) is the minimal possible number of critical points of a

Morse function S^{4}\backslash K\rightarrow S^{1} belonging to the canonical class in H^{1}(S^{4}\backslash K).

Question 8.1 (H. Endo, A. Pajitnov [4]). Is it true that for any 2‐knot K there
exists a strongly minimal Morse junction S^{4}\backslash K\rightarrow S^{1l}?
This is true for spun knots K=S(k) where k is a classical knot with \mathcal{M}\mathcal{N}(k)=2.

Question 8.2 (H. Endo, A. Pajitnov). Is it true that for any classical knot k wehave \mathcal{M}\mathcal{N}(S(k))=2\mathcal{M}\mathcal{N}(k)'?

This is true for any classical knot k of tunnel number 1,
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It is known [4] that \mathcal{M}\mathcal{N}(K_{1}\# K_{2})\leq \mathcal{M}\mathcal{N}(K_{1})+\mathcal{M}\mathcal{N}(K_{2}) for knots K_{1}, K_{2} ofany dimension.

Question 8.3 (H. Endo, A. Pajitnov). Is it true that\mathcal{M}\mathcal{N}(K_{1}\# K_{2})=\mathcal{M}\mathcal{N}(K_{1})+\mathcal{M}\mathcal{N}(K_{2})for2‐knots ?
i

Problem 8.4

surface‐links(H. Endo, A. Pajitnov). Compute Morse‐Novikov numbers for the9_{1)} 9_{1}^{0,1}, 10_{2}^{0,1}, 10_{1}^{1,1}of the Yoshikawas table [39J.
References[1] Abe, T., Tange, M., A construction of slice knots via annulus twists, arXiv:1305.7492.
[2] Ashihara, S., Calculating the fundamental biquandles of surface links from their ch‐diagrams,
J. Knot Theory Ramifications 21 (2012) 1250102 (23 pages).
[3] Ellwood, D. A., Ozsváth, P. S., Stipsicz, A. I., Szabó, Z. (ed), Floer homology, gauge theory,and low‐dimensional topology, Proceedings of the Clay Mathematics Institute Summer School

held in

Budapest, June, 2004. Clay Mathematics Proceedings 5. American MathematicalSociety, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2006.

[4] Endo, H., Pajitnov, A., On the Morse‐Novikov number for 2‐knots, arXiv:1502.06352.
[5] Fukaya, K., Gauge riron to topology (Gauge theory and topology) (in Japanese), Springer,
Tokyo, 1995.

[6]

—,

Symplectic kika (Symplective geometry) (in Japanese), Iwanami Shoten, 1999.

[7] Fukaya, K., Oh, Y.‐G., Ohta, H., Ono, K., Lagrangian intersection Floer theory: anomaly andobstruction. Part I, Part II, AMS/IP Studies in Advanced Mathematics 46.1, 46.2. American
Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009.

[8]

—,

Spectral invariants with bulk, quasimorphisms and Lagrangian Floer theory, arXiv:

1105.5123.

[9] Goda, H., Matsuda, H., and Pajitnov, A. Morse‐Novikov theory, Heegaard splittings, andclosed orbits of gradient flows, Algebra i Analiz 26 (2014) 131−158; reprinted in St. Petersburg
Math. J. 26 (2015) 441‐461.

[10] Hartley, R., Knots with free period, Canad. J. Math. 33 (1981) 91‐102.
[11] Harvey, W. J., Boundary structure of the modular group, Riemann surfaces and related topics:
Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y.,
1978), pp. 245‐251, Ann. of Math. Stud. 97, Princeton Univ. Press, Princeton, N.J., 1981.

[12] Hedden, M., Knot Floer homology of Whitehead doubles, Geom. Topol. 11 (2007) 101‐999.
[13] Joung, Y., Kamada, S., Lee, S. Y., Applying Lipsons state models to marked graph diagramsof surface‐links, J. Knot Theory Ramifications 24 (2015) 1540003 (18 pages).

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128

[14] Joung, Y., Kamada, S., Kawauchi, A., Lee, S. Y., Polynomial of an oriented surface‐link
diagram via quantum A_{2} invariant, arXiv:1602.01558.

[15] Joung, Y., Kim, J., Lee, S. Y., Ideal coset invariants for surface‐links in \mathbb{R}^{4}Ramifications 22 (2013) 1350052 (25 pages).

[16] Kamada, S., Kawamura, K.,

On,

J. Knot Theory

ribbon‐clasp surface‐links and normal forms of singularsurface‐links, arXiv: 1602.07855.

[17] Kamada, S., Kim, J., Lee, S. Y., Computations of quandle cocycle invariants of surface‐links
using marked graph diagrams, J. Knot Theory Ramifications 24 (2015) 1540010 (35 pages).
[18] Kawauchi, A., A chord diagram of a ribbon surface‐link, J. Knot Theory Ramifications 24(2015) 1540002 (24 pages).
[19] Kawauchi, A., Shibuya, T., Suzuki, S., Descriptions on surfaces in four‐space, I; Normal
forms, Math. Sem. Notes Kobe Univ. 10 (1982) 75‐125.

[20] Kim, J., Joung, Y., Lee, S. Y.,

On the Alexander biquandlesof oriented surface‐links via

marked graph diagrams, J. Knot Theory Ramifications 23 (2014) 1460007 (26 pages).

[21]

—,

On generating sets of Yoshikawa moves for marked graph diagrams of surface‐links,

J. Knot Theory Ramifications 24 (2015) 1550018 (21 pages).

[22] Kearton, C., Kurlin, V.,

All 2‐dimensional links in 4‐space live insideauniversal 3‐

dimensional polyhedron, Algebraic & Geometric Topology 8 (2008) 1223‐1247.

[23] Lee, S. Y., Invariants of surface‐links in \mathbb{R}^{4} via skein relation, J. Knot Theory Ramifications
17 (2008) 439‐469.
[24]

—,

Towards invariants of surfaces in 4‐space via classical link invariants, Trans. Amer.

Math. Soc. 361(2009) 237‐265.

[25] Lomonaco, S. J., Jr., The homotopy groups of knots I. How to compute the algebraic 2‐type,
Pacific J. Math. 95 (1981) 349‐390.
[26] Manolescu, C., Pin(2)‐equivariant Seiberg‐Witten Floer homology and the triangulation con‐jecture, J. Amer. Math. Soc. 29 (2016) 147‐176.

[27] Manolescu, C., Ozsváth, P., Thurston, D., Grid diagrams and Heegaard Floer invariants,
arXiv:0910.0078.

[28] Mark, T., Torsion, TQFT, and Seiberg‐Witten invariants of 3‐manifolds, Geom. Topol. 6(2002) 27‐58.
[29] Némethi, A., Graded roots and singularties, Singularities in geometry and topology, 394‐463,
World Sci. Publ., Hackensack, NJ, 2007.

[30] Ni, Y., Knot Floer homology detects fibred knots, Invent. Math. 170 (2007) 577‐608.

[31] Nozaki, Y.,An

explicit

relation between knot groups in lens spaces and those inS^{3},

arXiv: 1602.05884.

[32] Oh, Y.‐G., Lectures on Floer theory and spectral invariants of Hamiltonian flows, Morse the‐oretic methods in nonlinear analysis and in symplectic topology, 321‐416, NATO Sci. Ser. II

Math. Phys. Chem. 217, Springer, Dordrecht, 2006.

[33] Ozsváth, P. S., Stipsicz, A. I., Szabó, Z., A combinatorial description of the U^{2}=0 version
of Heegaard Floer homology, Int. Math. Res. Not. IMRN 2011, no. 23, 5412‐5448.
14

129

[34] Ozsváth, P. S., Szabó, Z., Heegaard Floer homology and alternating knots, Geom. Topol. 7(2003) 225‐254.
[35]

[36]

—,

—,

Holomorphic disks and genus bounds, Geom. Topol. 8 (2004) 311‐334.
Kauffman states, bordered algebras, and a bigraded knot invariant, arXiv:1603.06559.

[37] Sarkar, S., Wang, J., An algorithm for computing some Heegaard Floer homologies, Ann. of
Math. (2) 171 (2010) 1213‐1236.
[38] Swenton, F. J., On a calculus for 2‐knots and surfaces in 4‐space, J. Knot Theory Ramifications
10 (2001) 1133‐1141.
[39] Yoshikawa, K., An enumeration of surfaces in four‐space, Osaka J. Math. 31 (1994) 497‐522.
[40] Problem

list

Banff 2016,

Synchronisationofsmoothandtopological 4‐manifolds,Banff International Research Station, February 2016. http: / \mathrm{w}\mathrm{w}\mathrm{w} math. uqam. \mathrm{c}\mathrm{a}/\sim_{\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{l}1}/.

problem list‐banff.pdf

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