Problems on Low-dimensional Topology, 2016
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 18–20, 2016.

Contents
1 Heegaard Floer homology for embedded bipartite graphs

2

2 Heegaard Floer homology of knots and 3-manifolds

3

3 Modifying constructions of Lagrangian and Heegaard Floer theory

5

4 Braid groups and Cp -groups

6

5 Complex 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

11

8 Morse-Novikov numbers of surface-links

12

1 Research 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 16H02145 and 16K13754.

1

1Heegaard 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 Heegaard 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 version) 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.
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For a balanced bipartite graph GV1 ,V2 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
answer in 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
combinatorially? 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 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 complex?

2

2

Heegaard Floer homology of knots and 3-manifolds(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 Z/2Z is computable combinatorially [27].
Problem 2.1 (M. Tange). Find an algorithm to compute HF + (Y, s), HF − (Y, s)with Z coeﬃcients.
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 presentation, see [1]. Let An (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 An (K) are concordant or\
not. Hedden [12] gave a formula of CF
K of Whitehead double D+ (K, n) from thatof K. As an analogy, the following question is considered:
Problem 2.3 (M. Tange). Compute CF K ∞ (An (K)) from CF K ∞ (K).
In the case of K = 63 , the annulus twist of 63 has the same τ -invariant, i.e.,τ (63 ) = τ (An (63 )) holds for any n.
Question 2.4 (M. Tange). For any n does the equality τ (K) = τ (An (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 ′ equal?
For a knot K in S 3 and an integer n, let Sn3 (K) denote the 3-manifold obtained
from S 3 by n-surgery along K. A. Levine in [40] asks the following question:
If K1 is concordant to K2 , then for all n, Sn3 (K1 ) is homology cobordant
to Sn3 (K). Is the converse true?
Here we put some stronger problem:
Problem 2.6 (M. Tange). Find non-concordant knots K1 and K2 with same dinvariant for Sn3 (K1 ) and Sn3 (K2 ) for any integer n.
Due to [35] knot Floer homology detects the Seifert genus:{}\g(K) = max s HF
K(S 3 , K, s) ̸= 0 .
3

The τ -invariant gives the bound for 4-ball genus as follows:|τ (K)| ≤ g4 (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?
The knot Floer homology also detects fiberedness of a knot due to [30].
\
K is a fibered knot ⇔ HF
K(K, g(K)) ∼= Z.
\
What is a geometric property characterizing the knot Floer homology HF
K(K, i)for i < g(K)?
( def
Suppose that K is a lens space (or L-space) knot ⇔ ∃p ∈ Z such that Sp3 (K) is)\a lens space (or L-space) . Then any HF
K(K, i) is isomorphic to Z or {0}.
Question 2.8 (M. Tange). What is a geometric property characterizing the isomor\
phism HF
K(K, i) ∼= 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 → S 1 be a circle valued Morse function. Let τtop denote the torsionof cell complex of the infinite cyclic cover X̃ with respect to f . Let τMorse denotethe torsion of Morse complex of the infinite cyclic cover X̃ with respect to f . Morsecomplex is generated by the critical points of the Morse function and the diﬀerentials
are defined by counting of trajectories between critical points. Then Hutchings andLee proved the following formula:τMorse (t) · ζ(t) = τtop .
The diﬀerence ζ(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 → S 1 a circle valued Morsefunction. Let v be a half transversal flow for f and τv denote the Novikov torsion forv. Let R be the regular surface of f and h : R → R ‘a monodromy map’ generalized
to even any non-fibered knot. Let and ζh (t) the zeta function of dynamical systemh. Then the following equality holds:
τv (t) ζh (t) =∆K (t).
t−1

Problem 2.10 (M. Tange). Interpret this formula by using some Floer theory.
4

Heegaard Floer counterpart of the right-hand side is the Heegaard Floer homology
HF K − (K). This problem implies a decomposition of HF K − (K) into two somehomology theories for Novikov torsion and zeta function. The ζ-part would be
symplectic Floer homology for mapping classes.
Casson invariant is the first term of LMO invariant Z LM O (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 2.11 (M. Tange). By deforming Heegaard (or instanton) Floer theory insome sense, find the higher terms in it again.

3

Modifying constructions of Lagrangian and Heegaard Floer
theory(Kaoru Ono)
For a pair of Lagrangian submanifolds L1 , L2 in a closed symplectic manifold,
Lagrangian Floer homology HF∗ (L1 , L2 ) is defined from a chain complex generated
by intersection points of L1 and L2 whose diﬀerential counts pseudo-holomorphic()
disks; for details see [6, 7]. Further, in [7], it is extended to HF∗ (L1 , b1 ), (L2 , b2 )for “bounding cochains” bi of Li (i = 1, 2). For a 3-manifold M , the instanton Floer
homology of M is defined by “(infinite dimensional) Morse theory” for the ChernSimons 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?
Construction 1 (bounding cochain). For a topological space, (co)homology theory 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 (L1 , L2 ) be a (transversal)
pair of Lagrangian submanifolds in a closed (or ‘tame’) symplectic manifold (X, ω).
In general, Floer chain complex for (L1 , L2 ) is not defined. The obstruction isformulated in terms of filtered A∞ -algebras associated with Li , i = 1, 2. If MaurerCartan equations in these filtered A∞ -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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
induces a filtration on the complex. Using this filtration, one can obtain “essential 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 invariants [32] for Hamiltonian diﬀeomorphisms, torsion exponents in Lagrangian Floer(co)homology, etc.

4

Braid groups and Cp -groups(Yuta Nozaki)p
Let G be a group, and let p be a positiveAs{ integer.
} in{[31], we define C} (G) top
be the subgroup of G generated by the set g g ∈ G ∪ [g, h] g, h ∈ G , where[g, h] = ghg −1 h−1 . In other words, Cp (G) is the kernel of the natural projection
G → Gab /pGab , where Gab denotes the abelianization of G.
When there is a p-fold cyclic covering (S 3 , K) → (L(p, q), K ′ ) for a knot K in
3
S 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 Cp (π1 (L(p, q) \ K ′ )), since the following sequence is exact,

G(K) −→ π1 (L(p, q) \ K ′ ) −→ Z/pZ.
Therefore, for an arbitrary knot K, the following question naturally arises, which
was discussed in [31]. Here, as in [31], we call a group G a Cp -group if there exists
G′ such that G is isomorphic to Cp (G′ ).
Question 4.1 (Y. Nozaki). Let K be a knot. Is G(K) a Cp -group?
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 → L(p, q). We note that, if such a K ′ exists, G(K) is a Cp -group.
Hence, if we can show that G(K) is not a Cp -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 (Y. Nozaki). Let p be odd and n ≥ 3. Is the nth braid group Bn a
Cp -group?
6

It is known in [31] that, if G is a Cp -group and there is an epimorphism G →G′ whose kernel is a characteristic subgroup of G, then G′ is also an Cp -group.
Here, a characteristic subgroup of G is a subgroup which is invariant under all
automorphisms of G. Let Sn be the nth symmetric group. The kernel of thenatural homomorphism Bn → Sn is the pure braid group, which is a characteristic
subgroup of Bn . Since Sn is not a Cp -group for even p (as shown in [31, Example2.8]), it follows that Bn is not a Cp -group for even p. On the other hand, B3 is a
Cp -group for p with gcd(p, 6) = 1.
Remark. Let Xn be the configuration space of n distinct points in R2 . The braid
group Bn is isomorphic to π1 (Xn /Sn ). Problem 4.2 is related to a problem to find
an appropriate space whose p-fold cyclic cover is homeomorphic to Xn /Sn .

5Complex of surfaces of a 4-manifold and the adjunction
inequalities(Hokuto Konno)
The notion of the “complex of curves” of a surface was introduced by Harvey[11] in the 1980s, 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.
A 4-dimensional analog of this notion, namely, “complex of surfaces” was introduced by Mikio Furuta.2
Definition (M. Furuta) Let X be an oriented, closed smooth 4-manifold. The
complex of surfaces K = K(X) of X is the abstract simplicial complex defined asfollows:• The set of vertices V (K) is given as the set of smooth embeddings of surfaces
with self-intersection number zero:{}
V (K) := Σ ,→ X [Σ]2 = 0 .
Here we consider only oriented, closed, connected surfaces. We denote eachvertex (Σ ,→ X) ∈ V (K) briefly by Σ.
• For k ≥ 1, a collection of (k + 1) vertices Σ0 , . . . , Σk ∈ V (K) spans a k-simplex
if and only if Σ0 , . . . , Σk are disjoint.
In the above definition of the “complex of surfaces”, we do not consider the isotopy classes of embeddings of surfaces. On the other hand, in the same way as
the definition of the complex of curves, one can define an abstract simplicial complex whose vertices are the isotopy classes of embeddings of surfaces and whose
2 The author heard this definition from Mikio Furuta in private communication in April 2015.

7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 K(X) is contractible for any X. Thus it is natural toseek a significant subcomplex of K(X) having non-trivial homotopy type.
Definition (H. Konno) Let s be a spin c structure on X. Then, the complex of
surfaces violating the adjunction inequality KV = KV (X, s) is the subcomplex of
K(X) defined as the set of vertices is given by{}
V (KV ) := Σ ∈ V (K) max{−χ(Σ), 0} < |c1 (s) · [Σ]|and having the induced structure of an abstract simplicial complex from K.
We showed that, for any k ≥ 0, there exists infinitely many pairs (X, s) satisfyingH̃k (KV (X, s); Z) ̸= 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 H̃k (KV (X, s); Z) =0 except for the case of V (KV (X, s)) = ∅.
Problem 5.1 (H. Konno). Find an example of (X, s) and k with H̃k (KV (X, s); Z) =0 and V (KV (X, s)) ̸= ∅.
The complex of curves is used to describe the end of the moduli space of complex
structures on the base surface. On the other hand, the complex KV is used todescribe “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.
Problem 5.2 (H. Konno). For a closed 4-manifold, construct the “moduli space” ofa certain 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.

6

Surface-links and marked graph diagrams(Sang Youl Lee)A surface-link is a closed 2-manifold smoothly (or piecewise linearly and locallyflatly) embedded in R4 . Two surface-links are said to be equivalent if they areambient isotopic.
A marked graph diagram (or ch-diagram) is a ?link diagram in R2 possibly with? 
some 4-valent vertices equipped with markers: ???? . An oriented marked graph


8

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diagram is a marked graph diagram in whichedge has an orientation such?? every?? ⌞?? 
⌟?
that each marked vertex looks like ⌜??? or ??⌝? (see Figure 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 ???? with


and

?

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

said to be admissible if both resolutions L− (D) and L+ (D) are diagrams of (oriented)
trivial links.

L _ (D)

D

L +(D)

D

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 L is represented by an admissible marked graph diagram D. Moreover, if D is an admissible marked graph diagram representing a surface-link L, then one can construct asurface-link LD from D in a canonical way such that LD is equivalent to L.
Γ1 :?
⌜?⌝


Γ′1 :

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⌞?⌟

Γ2 :

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:: ⌝:: ::⌜  ::⌟:::: ⌞
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?  ??⌞? ??⌟
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⌝⌟??? ?⌞⌞? ?⌟Figure 2: Yoshikawa moves of type I

The Yoshikawa moves for oriented marked graph diagrams are the local moves
9

Γ6 :

⌜ ???

?
⌝

Γ′6 :

⌜ ??
?
?
⌝

/
o

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99  77 ⌜999⌞ ⌜77⌞7:: 99 ⌝ 7:⌜: 99 
:: 9 ⌞9 9⌟99

 

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⌜ ???⌞
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???
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?
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99  77 ⌜99⌞ ⌜77⌞7
9
:: 999⌝ 7⌜::: 999: 9
 ::: 9⌟99 ⌞Figure 3: Yoshikawa moves of type II

Γ1 , . . . , Γ5 (Type I) and Γ6 , . . . , Γ8 (Type II) illustrated in Figures 2 and 3. Let
S = {Γ1 , Γ′1 , Γ2 , Γ3 , Γ4 , Γ′4 , Γ5 , Γ6 , Γ′6 , Γ7 , Γ8 }.
It is known that two admissible marked graph diagrams represent equivalent surfacelinks if and only if they are transformed into each other by a finite sequence of
11 Yoshikawa moves in S [22, 20, 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 S. For unoriented surface-links(i.e., non-orientable surface-links or orientable surface-links without orientations),
the Yoshikawa moves in S forgetting the orientations are enough to describe theirmarked graph representations [19, 21, 22, 38].
On the other hand, it is proved that if Γ ∈ S − {Γ5 , Γ8 }, then Γ is independent
from the other moves in S [21]. If the answers of the following two questions are allaﬃrmative, then 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 Γ5 independent from the other moves in S?
Question 6.2 (J. Kim, Y. Joung, S. Y. Lee [21]). Is the Yoshikawa move Γ8 independent from the other moves in S?
Let L be a surface-link and let D be a marked graph diagram of L. Let |V (D)| and|C(D)| denote the number of all marked vertices and classical crossings in D, respectively. In [39], Yoshikawa introduced the ch-index, denoted by ch(L), of a surface-link
L, which is defined to be the minimum number ch(L) = minD∈D (|V (D)| + |C(D)|),
where D denotes the set of all marked graph diagrams representing L. Clearly, ch(L)is an ambient isotopy invariant of L. Using the terminology, he gave a table of 23
surface-links with ch-index ≤ 10 [39]. So it is natural to raise the following problem.
Problem 6.3 (S. Y. Lee). Create a complete table of admissible marked graph diagrams representing surface-links with ch-index ≥ 11.
10

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 surfacelink 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 (S. Y. Lee). Construct polynomial invariants for surface-links withmarked graph diagrams 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 surfacelinks with marked graph diagrams?
Question 6.8 (S. Y. Lee). Is it possible to construct a surface-link (co)homology
with marked graph diagrams.

7

Surface-links which bound immersed handlebodies(Kengo Kawamura)An immersed surface-link or simply a surface-link means a closed oriented surface generically immersed in R4 . When it is embedded, we also call it an embedded
surface-link. Two surface-links are equivalent if there is an orientation-preserving
diﬀeomorphism f : R4 → R4 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 R4 whose multiple point set consists of ribbon singularities. (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 R4 whose multiple point set consists of ribbon singularities andclasp singularities; for details, see [16].

11A chord graph (O; α) is a spatial trivalent graph which consists of a trivial link O
and disjoint simple arcs α spanning O. A chord diagram C(O; α) is a diagram of achord graph (O; α). It is known [18] that every ribbon surface-link can be obtained
from a chord graph (O; α) up to equivalence. The resulting ribbon surface-link,denoted by F (O; α), is obtained from a trivial 2-link whose equator is O by 1-handle
surgeries along 1-handles h(α) whose cores are α. A ribbon surface-link F (O; α)is faithfully equivalent to a ribbon surface-link F (O′ ; α′ ) if there is an equivalencef : R4 → R4 sending F (O; α) to F (O′ ; α′ ) and meridian curves of h(α) to nullhomotopic curves in F (O′ ; α′ ) ∪ h(α′ ). It is proved in [18] that two ribbon surfacelinks F (O; α) and F (O′ ; α′ ) are faithfully equivalent if and only if the chord diagrams
C(O; α) and C(O′ ; α′ ) are related by a finite sequence of certain moves.
We generalize above arguments as follows. A chord graph (O ∪ H; α) is a spatial
trivalent graph which consists of a split union of a trivial link O and Hopf links
H, and disjoint simple arcs α spanning O ∪ H. A chord diagram C(O ∪ H; α) isa diagram of a chord graph (O ∪ H; α). It can be seen that every ribbon-clasp
surface-link can be obtained from a chord graph (O ∪ H; α) up to equivalence. The
resulting ribbon-clasp surface-link, denoted by F (O ∪ H; α), is obtained from an
M -trivial 2-link (for details; see [16]) whose equator is O ∪ H by 1-handle surgeries
along 1-handles h(α) whose cores are α. We similarly define a faithful equivalence
for ribbon-clasp surface-links. Then, we ask whether an analogous result holds.
Problem 7.1 (K. Kawamura). Find certain moves for chord diagrams C(O ∪ H; α)which generate the faithful equivalence on ribbon-clasp surface-links F (O ∪ H; α).
This problem is a specialized version of the following problem.
Problem 7.2 (K. Kawamura). Find certain moves for chord diagrams C(O ∪ H; α)which generate the equivalence on ribbon-clasp surface-links F (O ∪ H; α).

8Morse-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 \K → S 1 on the complement to a 2-knot K is called strongly minimal if its numberof critical points mp (f ) of index p is minimal possible for every p. The MorseNovikov number MN (K) is the minimal possible number of critical points of a
Morse function S 4 \ K → S 1 belonging to the canonical class in H 1 (S 4 \ K).
Question 8.1 (H. Endo, A. Pajitnov [4]). Is it true that for any 2-knot K there
exists a strongly minimal Morse function S 4 \ K → S 1 ?
This is true for spun knots K = S(k) where k is a classical knot with MN (k) = 2.
Question 8.2 (H. Endo, A. Pajitnov). Is it true that for any classical knot k wehave MN (S(k)) = 2 MN (k)?
This is true for any classical knot k of tunnel number 1.
12

It is known [4] that MN (K1 #K2 ) ≤ MN (K1 ) + MN (K2 ) for knots K1 , K2 ofany dimension.
Question 8.3 (H. Endo, A. Pajitnov). Is it true that
MN (K1 #K2 ) = MN (K1 ) + MN (K2 )for 2-knots?
Problem 8.4 (H. Endo, A. Pajitnov). Compute Morse-Novikov numbers for thesurface-links
91 , 90,1
100,1
101,1
1 ,
2 ,
1
of the Yoshikawa’s table [39].

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).

13[14] Joung, Y., Kamada, S., Kawauchi, A., Lee, S. Y., Polynomial of an oriented surface-link
diagram via quantum A2 invariant, arXiv:1602.01558.
[15] Joung, Y., Kim, J., Lee, S. Y., Ideal coset invariants for surface-links in R4 , J. Knot Theory
Ramifications 22 (2013) 1350052 (25 pages).
[16] Kamada, S., Kawamura, K., On 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 biquandles of 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 inside a universal 3dimensional polyhedron, Algebraic & Geometric Topology 8 (2008) 1223–1247.
[23] Lee, S. Y., Invariants of surface-links in R4 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 conjecture, 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 singularities, 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 in S 3 ,arXiv:1602.05884.
[32] Oh, Y.-G., Lectures on Floer theory and spectral invariants of Hamiltonian flows, Morse theoretic 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[34] Ozsváth, P. S., Szabó, Z., Heegaard Floer homology and alternating knots, Geom. Topol. 7(2003) 225–254.
[35], Holomorphic disks and genus bounds, Geom. Topol. 8 (2004) 311–334.

[36], 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, “Synchronisation of smooth and topological 4-manifolds”,
Banﬀ International Research Station, February 2016. http://www.math.uqam.ca/~powell/
problem-list-banff.pdf

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