77

数理解析研究所講究録
第2004巻 2016年 77-88

Surface‐links and marked graph diagrams

Sang Youl Lee
Department of Mathematics, Pusan National University

1

Introduction

A surface‐link is a closed surface smoothly embedded in Euclidean 4‐space \mathbb{R}^{4}. A surface‐knot is a one component surface‐link.

A2‐sphere link is sometimes called a 2‐link. A2‐linkof one component is called a 2‐knot. Two surface‐links \mathcal{L} and \mathcal{L}' in \mathbb{R}^{4} are equivalent if

they are ambient isotopic, that is, there is an orientation preserving homeomorphismh : \mathbb{R}^{4}\rightar ow \mathbb{R}^{4} such that h(\mathcal{L})=\mathcal{L}' or, equivalently, there exists a smooth family ofdiffeomorphisms f_{s} : \mathbb{R}^{4}\rightarrow \mathbb{R}^{4}(s\in[0,1]) such that f_{0}=\mathrm{i}\mathrm{d}_{\mathbb{R}^{4} the identity of \mathbb{R}^{4} and
f_{1}(\mathcal{L})=\mathcal{L}' If each component \mathcal{K}_{i} of a surface‐link \mathcal{L}=\mathcal{K}_{1}\cup\cdots\cup \mathcal{K}_{ $\mu$}( $\mu$\geq 1) is oriented,
then \mathcal{L} is called an oriented surface‐link. Two oriented surface‐links \mathcal{L} and \mathcal{L}' are equivalent
if the restriction h|_{\mathcal{L} : \mathcal{L}\rightar ow \mathcal{L}' of h is also orientation preserving.
A marked graph diagram is a link diagram in \mathbb{R}^{2} possibly with some 4‐valent vertices
in which each 4‐valent vertex has a marker indicated by a small segment —
S. J.
K.
Jr.
and
Yoshikawa
introducedamethodofsurface‐links
Lomonaco,
presenting
[15]
[18]
using marked graph diagrams. Indeed, every surface‐link is presented by a marked graphdiagram (cf. [15, 18]) and such a presentation diagram is unique up to Yoshikawa moves(see Theorem 2.3). By using marked graph diagram presentation for surface‐links, someproperties and invariants of surface‐links were studied in [1, 2, 4, 6, 8, 9, 12, 13, 14, 16, 18].
In this short survey paper, we give a brief introduction to marked graph diagram presen‐tation of surface‐links and a method of constructing ideal coset invariants for surface‐links
introduced in [4, 14] by means of a polynomial invariant \ll\cdot\gg for marked graphs in \mathbb{R}^{3}defined by using a state‐sum model with classical link invariants as its state evaluation.
Section 2 presents marked graph diagram presentation of surface‐links. Section 3 deals
with the polynomial invariant \ll\cdot\gg for marked graphs in \mathbb{R}^{3} Section 4 discusses ideal
coset invariants derived from the polynomial \ll\cdot\gg An extended version of this paper
,

,

.

.

.

will be appear in elsewhere.

2

Marked graph diagrams of surface‐links

A marked graph is a spatial graph G in \mathbb{R}^{3} such that G is a finite regular graph possiblywith 4‐valent vertices, say v_{1}, v_{2} ,

.

.

.

,

v_{n} ; each v_{i} is a rigid vertex,

i.e., we fix a sufficiently

small rectangular neighborhood N_{i}\cong\{(x, y)\in \mathbb{R}^{2}|-1\leq x, y\leq 1\} , where v_{i} corresponds

to the origin and the edges incident to v_{i} are represented bywhich is the interval on N_{i} given byx^{2}=y^{2} ; each v_{i} has a marker,\displaystyle \{(x, 0)\in \mathbb{R}^{2}|-\frac{1}{2}\leq x\leq\frac{1}{2}\} Two marked graphs
.

are

78

equivalent if they are ambient isotopic in \mathbb{R}^{3} with keeping rectangular neighborhoods andmarkers.
An orientation of a marked graph G is a choice of an orientation for each edge of G

\geq $\xi$ \ovalbox{\t smal REJ CT}
Otherwise,

A marked graph is saidin such a way that every vertex in G looks likeor

to be orientable if it admits an orientation.

it is said to be nonorientable.

an

.

By

oriented marked graph we mean an orientable marked graph with a fixed orientation.

Two oriented marked graphs are equivalent if they are ambient isotopic in \mathbb{R}^{3} with keeping
rectangular neighborhoods, orientation and markers. An oriented marked graph G in \mathbb{R}^{3}
can be described as usual by a diagram D in \mathbb{R}^{2} which is an oriented link diagram in \mathbb{R}^{2}
possibly with some marked 4‐valent vertices whose incident four edges have orientations
illustrated as above, and is called an oriented marked graph diagram of G (cf. Figure 1).
,

Figure 1: Oriented marked graph diagrams and a nonorientable marked graph diagram
Two oriented marked graph diagrams in \mathbb{R}^{2} represent equivalent oriented marked
graphs in \mathbb{R}^{3} if and only if they are transformed into each other by a finite sequenceof oriented mark preserving rigid vertex 4‐regular spatial graph moves (simply, oriented
mark preserving RV4 moves) \mathrm{r}_{1}, $\Gamma$_{1}', $\Gamma$_{2}, $\Gamma$_{3}, $\Gamma$_{4}, $\Gamma$_{4}' and $\Gamma$_{5} shown in Figure 2, which consists

Yoshikawa moves of type I (see Theorem 2.3).
$\Gamma$_{1} :$\Gamma$ í :\vec{-}$\Gamma$_{2} :\vec{-}$\Gamma$_{3} :\leftarrow^{\rightarrow}$\Gamma$_{4} :\leftarrow^{\vec{}}$\Gamma$_{4}' :\backslash *_{-}t\leftarrow^{\vec{}}\supset\supset\mathfrak{D}\mathrm{C}/*-\sear ow\mathrm{t}_{\rangle}$\Gamma$_{5} :Figure 2: Oriented mark preserving RV4 moves

79

An unoriented marked graph diagram or, simply, a marked graph diagram is a nonori‐or an orientable but not oriented marked graphdiagram in \mathbb{R}^{2} Two marked
graph diagrams in \mathbb{R}^{2} represent equivalent marked graphs in \mathb {R}^{3} if and only if they aretransformed into each other by a finite sequence of the moves $\Omega$_{1}, $\Omega$_{2}, $\Omega$_{3}, $\Omega$_{4}, $\Omega$_{4}' and $\Omega$_{5)}entable.

where $\Omega$_{i} stands for the move $\Gamma$_{i} without orientation.

For an (oriented) marked graph diagram D , let L_{-}(D) and L_{+}(D) be the (oriented)
link diagrams obtained from D by replacing each marked vertex\timeswith) (and,respectively, as illustrated in Figure 3. We call L_{-}(D) and L_{+}(D) the negative resolution
and the positive resolution of \mathrm{D} respectively. An (oriented) marked graph diagram D isadmissible if both resolutions L_{-}(D) and L_{+}(D) are trivial link diagrams.
,

Figure 3: Marked graph diagrams and their resolutions
Let D be a given admissible marked graph diagram with marked vertices v_{1} ,

.

.

.

,

v_{n}.

Define a surface F(D)\subset \mathbb{R}^{3}\times[-1, 1] by

where \mathb {R}_{t}^{3}(\mathb{R}_t^{3},F(D)\cap mathb{R}_t^{3})=\left{\begin{ar y}{l (\mathb{R}^3,L_{+}(D)&\mathr {f}\mathr {o}\mathr {}0<t\leq1,\ (mathb{R}^3,L_{-}(D)\cup(bigcup_{i=1}^nB_{i})&\mathr {f}\mathr {o}\mathr {}t=0,\ (mathb{R}^3,L_{-}(D)&\mathr {f}\mathr {o}\mathr {}-1\leqt<0, \end{ar y}\ight.

:=\{(x_{1}, x_{2}, X3, x_{4})\in \mathbb{R}^{4}|x_{4}=t\} and B_{i}(1\leq i\leq n) is a band attached to
L_{-}(D) at each marked vertex v_{i} as illustrated in Figure 4. We call F(D) the proper
surface associated with D.

\rightarrow\displaystle\frac{\mathrm{B}_\mathrm{i}\mathrm{L}-(\mathrm{D})\cup\{\mathrm{B}_{\mathrm{i} \}Figure 4: A band attached to L_{-}(D) at v_{i}L_{-}(D) and L_{+}(D) have the orientations induced from the ori‐(cf. Figure 3), We assume that the proper surface F(D) is oriented sothat the induced orientation on L_{+}(D)=\partial F(D)\cap \mathbb{R}_{1}^{3} matches the orientation of L_{+}(D)When D is oriented,

entation of D.

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Since D is admissible, we can obtain a surface‐link from F(D) by attaching trivial disks in\mathbb{R}^{3}\times[1, \infty) and another trivial disks in \mathbb{R}^{3}\times(-\infty 1]. We denote the resulting (oriented),

surface‐link by \mathcal{L}(D) , and call it the(oriented) surface‐link associated with D.

It is wellknown that the isotopy type of \mathcal{L}(D) does not depend on the choices of trivial disks (cf.

Figure 5 shows a schematic picture of the surface‐link \mathcal{L}(D) associated with a[5, 7marked graph diagram D.

\mathrm{D}\mathrm{L}_{-}(\mathrm{D})\mathcal{L}(D)
Figure 5: A surface‐link \mathcal{L}(D) associated with a marked graph diagram D

Definition 2.1. Let \mathcal{L} be an (oriented) surface‐link in \mathb {R}^{4}byan(oriented)

markedsurface‐link \mathcal{L}(D) in \mathbb{R}^{4}.
Let D beanpresented by D.

graph diagramadmissible.

We say that \mathcal{L} is presented

D if \mathcal{L} is ambient

isotopic to the (oriented)(oriented) marked graph diagram. By definition, \mathcal{L}(D) is

From now on, we show that any (oriented) surface‐link is presented by an admissible(oriented) marked graph diagram. It is well known [7] that any surface‐link \mathcal{L} in \mathbb{R}^{4}=\mathbb{R}^{3}\times \mathbb{R} can be deformed into a surface‐link L' called a hyperbolic splitting of \mathcal{L} by anambient isotopy of \mathbb{R}^{4} in such a way that the projection p:\mathcal{L}'\rightarrow \mathbb{R} satisfies the followings:,,

all critical points are non‐degenerate,

all the index 0 critical points (minimal points) are in \mathbb{R}_{-1}^{3},
all the index 1 critical points (saddle points) are in \mathb {R}_{0}^{3},
all the index 2 critical points (maximal points) are in \mathb {R}_{1}^{3}.
Let \mathcal{L} be a surface‐link and let \mathcal{L}' be a hyperbolic splitting of \mathcal{L} Then the cross‐section.

\mathcal{L} Ó =\mathcal{L}is a spatial 4‐valent regular graph in \mathb {R}_{0}^{3}

\cap \mathb {R}_{0}^{3} at t=0

We give a marker at each 4‐valent vertex (saddle
point) that indicates how the saddle point opens up above as illustrated in Figure 6.
When \mathcal{L} is an oriented surface‐link, we choose an orientation for each edge of \mathcal{L} Ó sothat it coincides with the induced orientation on the boundary of \mathcal{L}'\cap \mathbb{R}^{3}\times(-\infty, 0 ] by.

81\timesFigure 6: A marker at a 4‐valent vertex

The resulting (oriented) marked
graph G := \mathcal{L} Ó is called an (oriented) marked graph presenting \mathcal{L} A diagram D of the(oriented) marked graph G is clearly admissible, and is called an (oriented) marked graphdiagram or (oriented) ch‐diagram presenting \mathcal{L} In conclusion, we state the followings.
the orientation of \mathcal{L}' inherited from the orientation of \mathcal{L}

.

.

.

Theorem 2.2([7]). (1) Let D be an admissible (oriented) marked graph diagram. Then

there is an (oriented) surface‐link \mathcal{L} presented by D.

(2) Let \mathcal{L} be an (oriented) surface‐link. Then there is an admissible (oriented) marked
graph diagram D presenting \mathcal{L}.

Theorem 2.3([9, 10, 17(1) Two oriented marked graph diagrams present the sameoriented surface‐link if and only if they are transformed into each other by a finite sequenceof oriented mark preserving RV4 moves in Figure 2, called oriented Yoshikawa moves oftype I and oriented Yoshikawa moves of type II in Figure 7.
,$\Gamma$_{6} :\underline{\rightarrow}\supset$\Gamma$_{6}' :\leftarrow^{\vec{}}\supset$\Gamma$_{7} :\vec{-}$\Gamma$_{8} :Figure 7: Oriented Yoshikawa moves of type II(2) Two unoriented marked graph diagrams present the same unoriented surface‐link

if and only if they are transformed into each other by a finite sequence of unoriented markpreserving RV4 moves $\Omega$_{1}, $\Omega$_{2}, $\Omega$_{3}, $\Omega$_{4}, $\Omega$_{4}', $\Omega$_{5} called unoriented Yoshikawa moves of type
I and unoriented Yoshikawa moves of type II $\Omega$_{6}, $\Omega$_{6)}'$\Omega$_{7} and $\Omega$_{8} where $\Omega$_{i} stands for themove $\Gamma$_{i} without orientation.
,

,

,

82

Polynomial invariants for marked graphs in \mathbb{R}^{3} via classical3

link invariants
Let R be a commutative ring with the additive identity 0 and the multiplicative identity
1 and let[ ] : {classical knots and links in \mathb {R}^{3} } \rightarrow R
be a regular or an ambient isotopy invariant such that for a unit $\alpha$\in R and an element$\delta$\in R,[_{/}^{\backslash }<)]= $\alpha$[)], [_{/}^{\backslash }[K\mathrm{O}]= $\delta$[K]

.

,

(3.1)

(3.2)

where K\mathrm{O} denotes any addition of a disjoint circle \mathrm{O} to a classical knot or link diagram
K.
For a given marked graph diagram D , letin R[x, y] defined by the following two rules:[[D]](x, y) ( [ \mathrm{D}] for short) be a polynomial(L1) [[D]]=[D] if D is a link diagram,(L2)[ \rangle\langle] =[\left\{\begin{ar ay}{l} \\ \end{ar ay}\right\}]x+[[)(] y.

When D is an oriented marked graph diagram and [links, then [[D]] is defined by the rules:] is an invariant for oriented(L1) [[D]]=[D] if D is an oriented link diagram,(L2)[ \S $\xi$] =[\left\{\begin{ar ay}{l} \aleph-\near ow\ \near ow\ap rox \end{ar ay}\right\}]x+[ ) $\zeta$] y)(L3)=[\left\{ begin{ar ay}{l} \backslash \rightar ow\ )= \end{ar ay}\right\}]_{X}+[ 2\backslash \upar ow] y.

Let D=D_{1}\cup\cdots\cup D_{m} be an oriented link diagram and let w(D_{i}) be the usual writheof the component D_{i}.

The self‐writhe sw(D) of D is defined to be the sumsw(D)=\displaystyle \sum_{i=1}^{m}w(D_{i})Now let D beamarked graph diagram.

component of L_{+}(D) and L_{-}(D).

.

We chooseanarbitrary orientation for each

When D is oriented, we choose orientations for L_{+}(D)and L_{-}(D) induced from the orientation of D.

We define the self‐writhe sw(D) of D by

sw(D)=\displaystyle \frac{sw(L_{+}(D) +sw(L_{-}(D) }{2},where sw(L_{+}(D)) and sw(L_{-}(D)) are independent of the choice of orientations because
the writhe of each component of L_{+}(D) and L_{-}(D) is independent of the choice of orien‐
tation for the component.

83

It is noted that the self‐writhe sw(D) is invariant under Yoshikawa moves except themove$\Omega$_{1} For $\Omega$_{1} and its mirror move, we have.

Definition 3.1. Let D be an (oriented) marked graph diagram. We define \ll D\gg(x, y)( \ll D\gg for short) to be the polynomial in variables x and y with coefficients in R givenby\ll D\gg=$\alpha$^{-sw(D)}[[D]](x, y)\in R[x, y].
Let D be an (oriented) marked diagram. A state of D is an assignment of T_{\infty} or T_{0} toeach marked vertex in D.

Let S(D) be the set of all states of D.

For each state $\sigma$\in \mathcal{S}(D) ,let D_{ $\sigma$} denote the (oriented) link diagram obtained from D by replacing marked vertices

of D with two trivial 2‐tangles according to the assignment T_{\infty} or T_{0} by the state $\sigma$ :\rangle\langle T_{\infty}\rightar ow, \rangle\langle T_{0}\rightar ow)(\cdot

3_{T_{\infty} $\xi$\rightar ow\ap rox\mapsto, \S_{T_{0} $\xi$\rightar ow \mathrm{j} $\zeta$,\geq\leqT_{\infty}\rightar ow\infty\mapsto 3_{T_{0} \leq\rightar ow)\upar ow.
Then \ll D\gg has the following state‐sum formula:\displaystyle \l D\g =$\alpha$^{-sw(D)}\sum_{ $\sigma$\in \mathcal{S}(D)}[D_{ $\sigma$}]x^{ $\sigma$(\infty)}y^{ $\sigma$(0)},where$\sigma$(\infty)

andrespectively.

$\sigma$(0) denote the numbers of the assignment T_{\infty} and T_{0} of the state $\sigma$,([14]). Let G be an (oriented) marked graph in \mathbb{R}^{3} and let D be an(oriented) marked graph diagram of G For any given regular or ambient isotopy invariant

Theorem 3.2.

[ ] : {classical (oriented) links in \mathb {R}^{3} } \rightarrow R
satisfying the properties (3.1) and (3.2), the polynomial \ll D\gg is an invariant for(oriented) Yoshikawa moves of type I, and therefore it is an invariant of the (oriented)
marked graph G in \mathbb{R}^{3}.
4

Ideal coset invariants for surface‐links

An oriented n ‐tangle

diagram (n\geq 1) is an oriented link diagram T in the rectangleI^{2}=[0, 1]\times[0 1 ] in \mathbb{R}^{2} such that T transversely intersect with (0,1)\times\{0\} and (0,1)\times\{1\},in n distinct points, respectively, called theendpoints of T.

84

Let T_{3}^{\mathrm{o}\mathrm{r}\mathrm{i} and T_{4}^{\mathrm{o}\mathrm{r}\mathrm{i} denote the set of all oriented 3‐ and 4‐tangle diagrams such that theorientations of the arcs of the tangles intersecting the boundary of I^{2} coincide with theorientations as shown in (a) and (b) of Figure 8, respectively.

(a)

(b)

Figure 8: Boundaries of 3, 4‐tangle diagrams
For U\in T_{3}^{\mathrm{o}\mathrm{r}\mathrm{i} and V\in T_{4}^{\mathrm{o}\mathrm{r}\mathrm{i} , let R(U) , R^{*}(U) , S(V) and S^{*}(V) denote the oriented link

diagrams obtained from the tangles U and V by closing as shown in Figures 9 and 10.

Figure 9: Closing operations R and R^{*} of a 3‐tangle U\mathrm{S}(\mathrm{V}) \mathrm{S}^{*}(\mathrm{V})Figure 10: Closing operations S and S^{*} of a 4‐tangle V

T_{3} and T_{4} denote the set of all 3‐ and 4‐tangle diagrams without orientations,
respectively. For U\in T_{3} and V\in T_{4} let R(U) R^{*}(U) S(V) and S^{*}(V) be the link
diagrams obtained by the same way as above forgetting orientations.
Let

,

,

,

85

([4]). For any given regular or ambient isotopy invariant

Definition 4.1[ ] : {classical (oriented) links in \mathb {R}^{3} } \rightarrow R
satisfying the properties (3.1) and (3.2), the [ ] obstruction ideal (or simply, [ ] ideal) I

is defined to be the ideal of R[x, y] generated by the polynomials in R[x, y] :P_{1}= $\delta$ x+y-1,

P_{2}=x+ $\delta$ y-1,

P_{U}=([R(U)]-[R^{*}(U)])xy, U\in T_{3}(T_{3}^{\mathrm{o}\mathrm{r}\mathrm{i}})P_{V}=([S(V)]-[S^{*}(V)])xy, V\in T_{3}(T_{4}^{\mathrm{o}\mathrm{r}\mathrm{i}})Theorem 4.2,.

([4]). The map:{(oriented) marked graph diagrams} \rightarrow R[x, y]/I

defined by\overline{[]}(D)=\overline{[D]}:=\ll D\gg+I

for any (oriented) marked graph diagram D is an invariant for (oriented) surface‐links.
Remark 4.3. Let F be an extension field of R.

By Hilbert Basis Theorem, the [ ] ideal

I is completely determined by a finite number of polynomials in F[x, y] , say p_{1}, p_{2} ,i.e., I=<p_{1}, p_{2}

,

.

.

.

,

.

.

.

,

p_{r},

p_{r}>.

In the rest of the paper,

wegive the ideals of Kauffman bracket for unoriented links

and Kuperbergs quantum A_{2} bracket for tangled trivalent graphs [11] and corresponding

ideal coset invariants for unoriented surface‐links and oriented surface‐links, respectively.
For more details, we refer to [3, 4, 14].

Let K be a knot or link diagram. The Kauffman bracket of K is a Laurent polynomial\langle K\}=\{K\rangle(A)\in R=\mathbb{Z}[A, A^{-1}] defined by the following rules:(B1) \langle \mathrm{O})=1,(B2) \{\mathrm{O}K'\rangle= $\delta$\langle K'\rangle(B3),$\delta$=-A^{2}-A^{-2},where\{/\backslash _{\backslash }\rangle=A\langle)(\rangle+A^{-1}\langle\rangle,

where \mathrm{O}K' denotes any addition of a disjoint circle \mathrm{O} to a knot or link diagram K'.
Note that the Kauffman bracket polynomial is invariant under Reidemeister moves except

the move $\Omega$_{1} and for $\alpha$=-A^{3}

,

we

have

\langle\grave{/}0\rangle= $\alpha$\langle)\rangle, \langle\infty_{/}\rangle=$\alpha$^{-1}\langle)\rangle.
Then the polynomial \ll D\gg=\ll D\gg(A, x, y) in Definition 3.1 is given by\ll D\gg=(-A^{3})^{-sw(D)}[[D]](A, x, y)=(-A^{3})^{-sw(D)}\displaystyle\sum_{$\sigma$\in\mathcal{S}(D)}x^{$\sigma$(\infty)}y^{$\sigma$(0)}\langleD_{$\sigma$}\rangle.

86

Theorem 4.4. The Kauffman bracket ideal I is the ideal of \mathbb{Z}[A, A^{-1}, x, y] generated by(-A^{2}-A^{-2})x+y-1,
x+(-A^{2}-A^{-2})y-1,(A^{8}+A^{4}+1)xy.
Moreover,the map\overline{\langle D\}}=\ll D\gg+I:{marked graph diagrams} \rightarrow \mathbb{Z}[A, A^{-1}, x, y]/I defined byfor any marked graph diagram D isaninvariant for unoriented

surface‐links.

For any given oriented marked graph diagram D , let \ll D\gg denote the polynomial
in \mathbb{Z}[a, a^{-1}, x, y] defined by the following recursive rules:(1) \ll $\Theta$\gg=1.
(2) If D and D' are two oriented marked graph diagrams related by oriented Yoshikawa
moves $\Gamma$_{1} $\Gamma$ í, $\Gamma$_{2}, $\Gamma$_{3}, $\Gamma$_{4}, $\Gamma$_{4}' and $\Gamma$_{5}, then \ll D\gg=\ll D'\gg.
,,(3) \ll D\sqcup $\Theta$\gg=(a^{-6}+1+a^{6})\ll D\gg.

(4)\ll\S $\xi$\gg=x\ll\nearrow\approx\aleph_{-}/\gg+y\ll) $\zeta$\gg.

(5)a^{-9}\l \nearrow^{\nwarrow\backslash }\gg-a^{9}\l \backslash _{/}^{\nearrow}\gg=(a^{-3}-a^{3})\l \backslash ,1\gg.

Theorem 4.5. Let I be the ideal of \mathbb{Z}[a, a^{-1}, x, y] generated by(a^{-6}+1+a^{6})x+y-1,
x+(a^{-6}+1+a^{6})y-1,(a^{12}+1)(a^{6}+1)^{2}xy.

\overline{\{\rangle}_{A_{2} {oriented marked graph diagrams} \rightarrow \mathbb{Z}[a, a^{-1}, x, y]/I definedinvariant forby \overline{\langle D\rangle}_{A_{2}}=\ll D\gg+I for any oriented marked graph diagram D isThen the map:anoriented surface‐links.

We remark that the ideal I of \mathbb{Z}[a, a^{-1}, x, y] in Theorem 4.5 is actually the ideal of Ku‐
perbergs quantum A_{2} bracket for oriented links and the map \overline{\{\rangle}_{A_{2} is the corresponding
ideal coset invariant for oriented surface‐links (cf.

following:[3, 11

We close this section with theQuestion 4.6. Is there a classical link invariant [ ] such that the [ ] ideal is trivial?

Acknowlegement
This work was supported by Basic Science Research Program through the National Re‐(NRF) funded by the Ministry of Education, Science andTechnology (2013\mathrm{R}\mathrm{I}\mathrm{A}\mathrm{I}\mathrm{A}2012446)
search Foundation of Korea.

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Department of Mathematics
Pusan National University
Busan 46241

KOREA\mathrm{E} ‐mail address:

sangyoul@pusan.ac.kr