Problems on Low-dimensional Topology, 2016 (Intelligence of Low-dimensional Topology)

Problems

on

Low‐dimensional

Topology,

2016

Edited

_by

T.

Ohtsukil

This is a list of_open

problems

onlow‐dimensional

topology

with

_expositions

of

their

_{history, background, significance,}

or

importance.

This listwasmade

by editing

manuscripts

written

_by

contributors ofopen

problems

tothe

_problem

sessionof the conference $\zeta$

(Intelligence

of Low‐dimensional

_Topology”’

held at Research Institute

for 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 \mathrm{C}^{p}‐groups 6

5

_Complex

of surfaces ofa4‐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

1ResearchInstitute forMathematicalSciences, KyotoUniversity,Sakyo‐ku,Kyoto, 606‐8502,JAPAN Email: [email protected]‐u.ac.jp

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

andAlexander

_grading

canbe calculated in an

explicit

and

combinatorial _way; and as a

corollary

(3)

the

Heegaard

Floer

homology

(hat

ver‐

sion)

is

_explicitly

determined

_by

the

signature

and theAlexander

_polynomial

of the

given

link. Howeverin

_general,

acombinatorial

description

of the differential of the

complex

is unknown.

\swarrow\leftrightarrow\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

forit from its

_diagram

in

S^{2}

with a

basepoint.

We also_prove that

(1)

the

generators

of the

_Heegaard

Floer chain

_complex

for the

_graph

are

_given

by

thestates of the

_diagram

_{(Figure above); (2)}

the Alexander

_polynomial

of the

_graph

can be

expressed

as a statesum. We _mayask the

following

questions,

whichwe

_suggest

to

answerin order.

Question

1.1

_(Y.

_Bao).

Can any two statesbe connected

by

transpositions

of

type

I

andII’? Is it

_possible

tocalculate the

_(relative)

Maslov

_grading

andAlexander

_grading

combinatorially9

For

_{“alternating”}

_(there

isnostandard

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 a

bigraded

chain

_complex

over

\mathbb{F}[U]

,the

homology

of which is showntobe

isomorphic

tothe knot Floer

_homology

_{(minus version).}

Itis

_{freely generated by}

theKauffman

states,

and itsdifferential is defined

_{algebraically,}

builtonbordered Floer

homology.

Wemay ask the

following

question.

Question

1.2

_{(Y. Bao).}

For a

bipartite graph,

or a

general graph,

is it

possible

to

2

_Heegaard

Floer

_homology

of knots and 3‐manifolds

(Motoo

Tange)

The hatversionof

_Heegaard

Floer

_homology

is

_{algorithmically computable}

_using

the

_{Heegaard diagram.}

This result isdue to

_[37].

Another

_development

of this

topic

isthe 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].

Problem 2.1

_{(M. Tange).}

Find an

algorithm

to

_compute

_{HF^{+}(Y, \mathfrak{s})}

,

HF^{-}(Y, \mathfrak{s})

with \mathbb{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}

‐bundleover a

surface.

Problem 2.2

_{(M. Tange).}

Find d ‐invariant

_{formula for hyperbolic 3‐manifold.}

Let K be aknot with annulus

presentation.

For the definition of annulus _pre‐

sentation,

see

[1].

Let

A^{n}(K)

be then‐fold annulus twist of K

along

the

embedding

annulus. The annulus twist is useful methodto construct a

pair

of the

diffeomorphic

0_‐surgeryor

diffeomorphic

0‐framed

_{2‐handlebody.}

Iftwoknots 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

not. Hedden

_[12]

_gave a formula of CFK of Whitehead double

D_{+}(K, n)

from that

of K. As an

analogy,

the

following

question

is considered:

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 the same $\tau$

‐invariant,

i. e.,

$\tau$(6_{3})= $\tau$(A^{n}(6_{3}))

holds for _anyn.

Question

2.4

_{(M. Tange).}

For _anyn does the

equality

$\tau$(K)= $\tau$(A^{n}(K))

hold,

in

general!?

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 aknot 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 concordantto

_{K_{2_{J}}}

then

_for

alln,

S_{n}^{3}(K_{1})

is

homology

cobordant

to

_{S_{n}^{3}(K)}

. Is the converse tru

e^{\ell}

Here we

_put

some

_stronger

problem:

Problem 2.6

_{(M. Tange).}

Find non‐concordant knots

_{K_{1}}

and

K_{2}

with same d‐

invariant

_for

_{S_{n}^{3}(K_{1})}

and

_{S_{n}^{3}(K_{2})}

_for

_any

_integer

n.

Dueto

_[35]

knot Floer

_homology

detects the Seifert _genus:

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 isnot known.

Question

2.7

_{(M. Tange).}

Does knot Floer

_homology

detect the

_4‐ball

_genus9

The knot Floer

_homology

also detects fiberedness ofaknot dueto

[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

\overline{HFK}(K, i)

for

_i<g(K)

?

Suppose

that K is alens _space

(or

\mathrm{L}

‐space)

knot

(^{\mathrm{d}}\Leftrightarrow^{\mathrm{e}\mathrm{f}}\exists p\in \mathbb{Z}

such that

S_{p}^{3}(K)

is a lens _space

(or

\mathrm{L}

‐space)).

Then_any

\overline{HFK}(K, i)

is

isomorphic

to \mathbb{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 weraised the

following problem.

Problem2.9

_{(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 torsion

of cell

_complex

of the infinite

_cyclic

cover

\tilde{X}

with

_respect

to

f

. Let $\tau$_Morse denote

the torsionof 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

and

Lee

_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 alevel set. The

right‐hand

side is

equivalent

to a summation

of

_{Seiberg‐Witten}

invariants on X dueto the result

_[28]

_by

Mark.

Asa

Hutchings‐Lee

_type

formula, Goda, Matsuda,

and

Pajitnov

in

[9]

proved

the

formula below. Let K be a knot in

S^{3}

and

f

:

S^{3}-K\rightarrow S^{1}

a circle valued Morse

function. Let vbeahalf transversal flow for

f

and_{$\tau$_{v}} denote the Novikov torsionfor v. Let Rbe the

regular

surface of

f

and h : R\rightarrow R ‘a

monodromy map’

generalized

to even _anynon‐fiUered knot. Let and

$\zeta$_{h}(t)

thezeta function of

_dynamical

system

h. Then the

following equality

holds:

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

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 some

homology

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

geometric

meanings.

Here we _propose the

following problem.

Problem 2.11

_{(M. Tange).}

_By

_{deforming Heegaard}

_{(or instanton)}

Floer

_theory

in

some _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

L_{1}, L_{2}

in a closed

symplectic

manifold,

Lagrangian

Floer

_homology

_{HF_{*}(L_{1}, L_{2})}

isdefined fromachain

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)}

. Fora3‐manifold M_,the instanton Floer

homology

of M is defined

_by

( (

(infinite

dimensional)

Morse

_theory”

for the Chern‐ Simons functional onthe _space of

SU(2)

connections onM_; see _e.g.

[5]

. Motivated

by

the instanton Floer

_homology,

for a3‐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

them to low dimensional

_topology

l_?

Construction 1

_(bounding

_cochain).

For a

topological

_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 ina 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

submanifoldsin aclosed

(or (tame’)

symplectic

manifold

(X, $\omega$)

.

In

_general,

Floer chain

_complex

for

_{(L_{1}, L_{2})}

is not defined. The obstruction is

formulated interms of filtered

A_{\infty}

‐algebras

associated with

_{L_{i},}

i=1,2. If Maurer‐

Cartan

_equations

inthesefiltered

_{A_{\infty}‐algebras}

have solutions

_(we

call them

_bounding

cochainsorMaurer‐Cartan

elements),

we can

modify

thedefinition 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 so

isomorphic.

The notion of

_augmentation

in the

_setting

ofcontact

_homology

is an

analog

of

_bounding

cochains.

Thereisalsoa

construction,

called bulk deformations

[8].

In

fact,

a

special

kind of

bulk deformations had

_already

usedin

_Heegaard

Floer

_theory

from the

_early

stage.

Construction 2

_(filtration

_by

the action

_functional).

Floer

_complex

is a kind of

Morse‐Novikov

_complex

associatedto 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 “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

(co)homology,

etc.

4

Braid

_groups

and

\mathrm{C}^{p}

‐groups

(Yuta Nozaki)

Let G be a_group, and let_pbe a

positive integer.

As in

[31],

we define

\mathrm{C}^{p}(G)

to be the

_subgroup

of G

_{generated by}

theset

\{g^{p}|g\in G\}\cup\{[g, h]|g, h\in G\}

, where

[g, h]=ghg^{-1}h^{-1}

. In other

words,

\mathrm{C}^{p}(G)

is the kernel of the natural

projection

G\rightarrow 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 shownin

[31]

that the knot group

G(K)

is

_isomorphic

to

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

since the

_following

sequence isexact,

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

was discussed in

[31].

_Here,

asin

[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')}

.

Question

4.1

_(Y.

_Nozaki).

Let K be a knot. Is

G(K)

a

\mathrm{C}^{p}-group2

Remark. For a

_given

knot Kin

S^{3}

,it is a non‐trivial

problem

to determine whether

there is aknot 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,

ifwe can show that

G(K)

isnot a \mathrm{C}^{p}‐group, it follows that such a K' does

not exist.

Remark.

_Hartley

_[10]

_gave alist of

possible

free

period

of

prime

knots K with_{up to}

10

_crossings.

Sucha K canbe obtained as the

preimage

ofaknot in

L(p, q)

.

Further,

we consider the

corresponding problem

for braid _groups.

Problem 4.2

_(Y.

_Nozaki).

Letp be odd and

n\geq 3

. Is the nth braid group

B_{n}

a

It is known in

_[31]

_that,

if G is a \mathrm{C}^{p}_‐group and there is an

epimorphism

G\rightarrow

G' whose kernel is a 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 the

natural

_homomorphism

_{B_{n}\rightarrow \mathfrak{S}_{n}}

isthe pure braidgroup, which is a characteristic

subgroup

of

B_{n}

. Since

\mathfrak{S}_{n}

is not a \mathrm{C}^{p}‐group for evenp

(as

shown in

[31,

Example

2.8]),

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 forpwith

\mathrm{g}\mathrm{c}\mathrm{d}(p, 6)=1.

Remark. Let

_{X_{n}}

be the

_{configuration}

space ofn 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

_appropriate

_spacewhose_p‐fold

_cyclic

cover is

homeomorphic

to

_{X_{n}/\mathfrak{S}_{n}.}

5

_Complex

of surfaces of

a

4‐manifold and the

adjunction

inequalities

(Hokuto Konno)

The notion of the $\zeta$

‘complex

of curves”’ of a surface was introduced

by Harvey

[11]

inthe 1980\mathrm{s}, and has been studied from the

viewpoint

of the Teichmüllerspace

and the action of the

_mapping

classgroup. The

complex of

curves

(also

called curve

complex)

ofa 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

are

spanned

by

collections of such curveswhich canbe 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 as

follows:

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, closed,

connected surfaces. We denote each

vertex

_{( $\Sigma$\mapsto X)\in V(\mathcal{K})}

_{briefly by}

$\Sigma$.

For

_{k\geq 1}

, acollection of

(k+1)

vertices

$\Sigma$_{0}

,.. . _,

$\Sigma$_{k}\in V(\mathcal{K})

spansa k

‐simplex

if and

_only

if

$\Sigma$_{0}

,. .. _,

$\Sigma$_{k}

are

disjoint.

In the above definition of the

_“complex

of

_surfaces”,

we do not consider the iso‐

topy

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

com‐

plex

whose vertices are the

isotopy

classes of

embeddings

of surfaces and whose

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 astheset 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 inducedstructureof 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

aclassical

application;

we can drive certain

adjunction inequalities

for surfaces embedded to 4‐manifolds whose

_{Seiberg‐Witten}

invariants vanish.

On the other

_hand,

wedonotknowany

example

of

(X,

s)

and k with

\tilde{H}_{k}(\mathcal{K}_{V}(X, \mathfrak{s});\mathbb{Z})=

0

_except

for the caseof

V(\mathcal{K}_{V}(X,\mathfrak{s}))=\emptyset.

Problem 5.1

_{(H. Konno).}

Findan

example of

(X, s)

andk with

\tilde{H}_{k}(\mathcal{K}_{V}(X, \mathfrak{s});\mathbb{Z})=

0 and

_{V(\mathcal{K}_{V}(X, \mathfrak{s}))\neq\emptyset.}

The

_complex

ofcurvesisusedto describe the end of the modulispace of

complex

structures on the base surface. On the other

hand,

the

complex

\mathcal{K}_{V}

is used to describe

_{‘(stretching”}

of

_{neighborhoods}

of embedded surfaces in 4‐manifold. The

Seiberg‐Witten equations

onthe stretched

neighborhoods play

a

key

roletoconsider

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).

Fora closed

4‐manifold,

constructthe “moduli

_{space” of}

acertainstructurewhose endis

_given

_by

_{4‐manifolds}

with

_cylindrical

ends. Describe the end

_of

the modulispace 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

locally

flatly)

embedded in

\mathbb{R}^{4}

. Two surface‐links are said to be

equivalent

if

they

are

ambient

isotopic.

A marked

_{graph diagram}

_{(or ch‐diagram)}

is a link

diagram

in

\mathbb{R}^{2}

possibly

with some 4‐valent vertices

equipped

with markers:

diagram

is a marked

graph diagram

in which _every

edge

has an orientation such

that each markedvertexlooks like

3

$\xi$

or

\geq\leq

(see

Figure

1).

Fora

given

(oriented)

marked

graph diagram

D,let

L_{-}(D)

and

L_{+}(D)

be classical

(oriented)

link

_diagrams

obtainedfrom D

_{by replacing}

each markedvertex

Xwith

) (

and ,

respectively

(see

Figure

1).

An

(oriented)

marked

graph diagram

D is

saidto\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}anditsresolutions

S. J.

_Lomonaco,

Jr.

_[25]

and K. Yoshikawa

_[39]

introducedamethod of

describing

surface‐links

_using

marked

_{graph diagrams. Indeed,}

_every surface‐link \mathcal{L} is repre‐ sented

_by

an admissible marked

graph diagram

D.

Moreover,

if D is an admissi‐

ble marked

_graph

_diagram

representing

a surface‐link \mathcal{L}, then one can construct a

surface‐link

\mathcal{L}_{D}

from D in a canonical_way such that

\mathcal{L}_{D}

is

equivalent

to\mathcal{L}.

$\Gamma$_{1} : $\Gamma$

_í

: $\Gamma$_{2} : $\Gamma$_{3} : $\Gamma$_{4}:

$\Gamma$_{4}'

:

\backslash \ovalbox{\tt\small REJECT}_{-}\nearrow'

$\Gamma$_{5}: \leftarrow^{\vec{}}

\supset

\leftarrow^{\vec{}}

\supset

\vec{\leftarrow}

\mathfrak{D}\mathrm{C}

\leftarrow^{\rightarrow} \leftarrow^{\vec{}} \leftarrow^{\vec{}}

/*-\aleph

Figure 2: Yoshikawamovesof_typeI

$\Gamma$_{6}: \leftarrow^{\rightarrow}

\supset

$\Gamma$_{6}'

: \vec{-}

\supset

$\Gamma$_{7}:

$\Gamma$_{8} : \leftarrow^{\rightarrow}

Figure3: Yoshikawamovesof_typeII

$\Gamma$_{1}

,...

,

$\Gamma$_{5}

(Type I)

and

$\Gamma$_{6}

,.. .

,

$\Gamma$_{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 isknown thattwoadmissible marked

_{graph diagrams}

_represent

_equivalent

surface‐ links if and

_only

if

_they

are transformed into each other

by

a finite _sequence of

11 Yoshikawa moves in

\mathfrak{S}[22

,

20,

38

]

. Therefore any oriented surface‐link can be

represented by

anoriented marked

graph diagram

[25, 39],

and sucha

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 movesin \mathfrak{S}

forgetting

the orientations are

enough

to describe their

marked

_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 othermovesin

\mathfrak{S}[21]

. If theanswers of the

following

two

questions

are all

affirmative,

then \mathfrak{S} is aminimal

generating

set for oriented Yoshikawamoves.

Question

6.1

_(J.

_Kim,

Y.

_Joung,

S. Y. Lee

_[21]).

Is the Yoshikawa move

$\Gamma$_{5}

inde‐

pendent

from

the othermoves in\mathfrak{S} ‘?

Question

6.2

_(J.

_Kim,

Y.

_Joung,

S. Y. Lee

_[21]).

Is the Yoshikawa move

$\Gamma$_{8}

inde‐

pendent

from

the othermoves in\mathfrak{S}^{9}

Let\mathcal{L} beasurface‐link and let D beamarked

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})}

,ofasurface‐link

\mathcal{L},which is defined tobe the minimum number

\displaystyle \mathrm{c}\mathrm{h}(\mathcal{L})=\min_{D\in \mathcal{D}}(|V(D)|+|C(D)|)

,

where \mathcal{D} denotes thesetof all marked

_{graph diagrams}

_representing

\mathcal{L}.

Clearly,

\mathrm{c}\mathrm{h}(\mathcal{L})

is an ambient

isotopy

invariant of \mathcal{L}.

Using

the

terminology,

he gave a table of 23

surface‐links with ch‐index

_{\leq 10[39]}

. Soit isnaturaltoraisethe

following problem.

Problem 6.3

_(S.

Y.

_Lee).

Create a

complete

table

of

admissible marked

graph

dia‐

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}

canbe considered.

Problem 6.4

_(S.

Y.

_Lee).

Howto

_compute

known invariants

_{for surface‐links}

_using

marked

_graph

_diagrams./’

In

_[25],

S. J.

_Lomonaco,

Jr. used marked

_{graph diagrams}

tocalculate thesurface‐ link _groups. In

_[2],

_{S. Ashihara gave} a method of

calculating

the fundamental

biquandles

of surface links from their marked

_{graph diagrams}

and Y.

_Joung,

J.

Kim andS. 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

invariantsfor surface‐links canbe

computed by

using

marked

graph diagrams

[17].

Theanswers of the

following problems

would enrich the

_theory

of surface‐links.

Problem 6.5

_(S.

Y.

_Lee).

Constructnew 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}

with marked

_{graph diagrams}

which can be

computed by

recursive rules

(skein relation)

and

_{categorifications.}

So

_far,

there have been several

attempts

to constructnewinvariantswith marked

graph diagrams

[13,

14, 15, 23,

24].

Finally,

one_may ask the

following

questions.

Question

6.7

_(S.

Y.

_Lee).

Isit

_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.}

7

Surface‐links which bound immersed handlebodies

(Kengo Kawamura)

An immersed

_{surface‐link}

or

simply

a

surface‐link

means a closed oriented sur‐

face

_generically

immersed in

\mathbb{R}^{4}

. When it is

embedded,

wealso call it an embedded

surface‐link. Two surface‐links are

equivalent

if there is an

orientation‐preserving

diffeomorphism f

:

\mathbb{R}^{4}\rightarrow \mathbb{R}^{4}

sending

oneto 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

setconsists ofribbon

_singulari‐

ties.

_(Note

that ribbon surface‐links are embedded

surface‐links.)

A surface‐link is

said 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}

and

A chord

_graph

_{(O; $\alpha$)}

is a

spatial

trivalent

graph

which consists ofatrivial link O

and

_{disjoint simple}

arcs $\alpha$

_spanning

O. A chord

_diagram

C(O; $\alpha$)

is a

diagram

ofa

chord

_graph

_{(O; $\alpha$)}

. It isknown

[18]

thateveryribbon surface‐linkcan be obtained

from a chord

graph

(O; $\alpha$)

_{up to}

equivalence.

The

resulting

ribbon

surface‐link,

denoted

_by

_{F(O; $\alpha$)}

, isobtained fromatrivial 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; $\alpha$)

is

_{faithfully equivalent}

to a ribbon surface‐link

F(O';$\alpha$')

if there is an

equivalence

f

:

\mathbb{R}^{4}\rightarrow \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]

thattwo ribbon surface‐

links

_{F(O; $\alpha$)}

and

_{F(O';$\alpha$')}

are

faithfully equivalent

if and

only

if the chord

diagrams

C(O; $\alpha$)

and

_{C(O';$\alpha$')}

arerelated

by

afinite _sequence of certainmoves.

We

_generalize

above

arguments

asfollows. A chord

graph

(O\cup H; $\alpha$)

isa

spatial

trivalent

_graph

which consists of a

split

union of a trivial link O and

Hopf

links

H, and

disjoint simple

arcs $\alpha$

spanning

O\cup H. A chord

diagram

C(O\cup H; $\alpha$)

is

a

diagram

of a chord

graph

(O\cup H; $\alpha$)

. It can be seen that every

ribbon‐clasp

surface‐linkcanbe obtained fromachord

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 whetheran

analogous

result holds.

Problem 7.1

_{(K. Kawamura).}

Find certainmoves

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).

Findcertain 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}

onthe

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

tothe 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 truefor spunknots

K=S(k)

where k is aclassical knot with

\mathcal{M}\mathcal{N}(k)=2.

Question

8.2

_(H.

_Endo,

A.

_Pajitnov).

Is it true that

_for

_any classical knot k we

have

_{\mathcal{M}\mathcal{N}(S(k))=2\mathcal{M}\mathcal{N}(k)'}

?

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}

of anydimension.

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})

for

2‐knotsi_?

Problem 8.4

_(H.

_Endo,

A.

_Pajitnov).

_Compute

Morse‐Novikov numbers

_for

the

surface‐links

9_{1)} 9_{1}^{0,1}, 10_{2}^{0,1}, 10_{1}^{1,1}

of

the Yoshikawa’s table

_[39J.

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