数理解析研究所講究録
第 1777 巻 2012 年 43-49

43Handlebody-links andHeegaard splittings of link complements
広島大学大学院理学研究科

張娼姫 (Yeonhee Jang)

Graduate School of Science,
Hiroshima University
In this article, we give an observation on certain relations between handlebodylinks and Heegaard splittings of link exteriors (or bridge presentations of links).

1. Handlebody-links
A spatial graph is a graph embedded in the 3-sphere . Two spatial graphs
are said to be equivalent if there exists an orientation-preserving homeomorphism:
which sends one of the spatial graphs to the other. It is known, by[14] and [24] for example, that two spatial graphs are equivalent if and only iftheir diagrams can be carried into each other by a finite sequence of elementary
moves in Figure 1.
$S^{3}$$f$$S^{3}arrow S^{3}$Figure 1:

A handlebody-link is a union of handlebodies embedded in the considered up
to ambient isotopy, namely, two handlebody-links are said to be equivalent if there
exists an orientation-preserving homeomorphism :which sends oneof the handlebody-links to the other. By considering the spines of handlebodies,
handlebody-links can be regarded as spatial graphs up to certain local movein a 3-ball, called IH-move, replacing an I-shaped part with an H-shaped part.
Throughout this article, we do not distinguish between handlebody-links andtheir spines. Ishii [5] proved that two handlebody-links are equivalent if and only
if their diagrams can be carried into each other by a finite sequence of elementary
moves in Figure 1 and Figure 2.
$S^{3}$$f$$S^{3}arrow S^{3}$

Remark 1 Let $K$ andbe two graphs embedded in . Apparently, $K$ and$K’$ are equivalentas handlebody-links if they are equivalent as spatial graphs.
$K^{l}$$S^{3}$

44

Figure 2:
However, the converse does not hold in general. For example, two spacial graphs
represented by the diagrams in Figure 3 are not equivalent (as spatial graphs)
while they are equivalent as handlebody-links.

$K’$

$K$

Figure 3:
Compared to the fact that various kinds of invariants are known for spacialgraphs, not many invariants are known for handlebody-links yet. Invariants ofhandlebody-links defined by using “quandles“ have been given in [6, 7, 13], andMizusawa and Murakami [16] introduced another invariant of handlebody-knots
by using certain quantum invariant for spatial graphs. Ishii and Masuoka [8]also introduced invariants of handlebody-knots each of which is defined from
unimodular Hopf algebra.

2. Unknotting tunnel systems of knots and Heegaard splittings of knot exteriors
For a knot $K$ in the 3-sphere , let $N(K)$ be the regular neighborhood of $K$ inby $E(K)$ , that is, $E(K)$ is the closure. We denote the exterior of $K$ in
embeddedofin . Let be the union of disjoint arcs
. We call an unknotting tunnel system of $K$ if theinso that,is a $genus-(n+1)$ handlebody, which we denote byclosure ofin
which is taken
where $N’(K\cup\tau)$ denotes a regular neighborhood of$N(K)$unknotting
tunnel ofparticular,
we call anto containin its interior. In$K$ when, is ais 1. Note that $N’(K\cup\tau)\cap E(K)$ , or the closure of$E(K)$and the closure
into“compression-body”. Thus the decomposition ofis a so-called $(genus-(n+1))$ Heegaard splitting of $E(K)$ .
ofWe say that two unknotting tunnel systems and ‘ of a knot $K$ are equivalent
if there exists an orientation-preserving homeomorphism : $E(K)arrow E(K)$ suchthat $f(\tau\cap E(K))=\tau’\cap E(K)$ . It can be easily seen that every Heegaard splittingof $E(K)$ can be obtained from an unknotting tunnel system of $K$ and that twounknotting tunnel systems of a knot $K$ are equivalent if and only if the Heegaard
splittings of $E(K)$ obtained from the unknotting tunnel systems are equivalent,$S^{3}$$S^{3}$$S^{3}$$S^{3}\backslash N(K)$$S^{3}$$t_{1},$$\tau$$K\cap\tau=\partial\tau$$S^{3}$$t_{2},$$\ldots,$$t_{n}$$\tau$$S^{3}\backslash N’(K\cup\tau)$$H_{\tau}$$K\cup\tau$$S^{3}$$\tau$$E(K)\backslash H_{\tau}$$n$$H_{\tau}$$E(K)\backslash H_{\tau}$$\tau$$\tau$$f$45
that is, there is an orientation-preserving homeomorphism of $E(K)$ carrying oneof the Heeagaard splittings to the other. Thus we do not distinguish between thetwo concepts, unknotting tunnel systems of knots and Heegaard splittings of knot
exteriors, in this article. Unknotting tunnels of knots or Heegaard splittings ofknot exteriors were studied by various mathematician (see [1, 2, 4, 15, 17, 18, 21]
for example),
When a knot $K$ in
and its unknotting tunnel system are given, we canobtain from them a handlebody-link $N(K)\cup H_{\tau}$ . We denote this handlebody-link
by $H(K, \tau)$ . Then we have the following proposition.
$S^{3}$$\tau$

Proposition 2 Let $K$ be a knot in , and let andbe unknotting tunnel
systems of K. If the two unknotting tunnel systems and are equivalent then$H(K, \tau)$ and $H(K, \tau’)$ are equivalent as handlebody-links.
$S^{3}$$\tau’$$\tau$$\tau’$$\tau$

Proof. Assume that and are equivalent. By the definition of equivalenceof unknotting tunnel systems, we can find an orientation-preserving homeomorphism : $E(K)arrow E(K)$ such that $f(\tau\cap E(K))=\tau’\cap E(K)$ . This implies that
sends $N’(K\cup\tau)\cap E(K)$ to $N’(K\cup\tau’)\cap E(K)$ , and hence $H(\tau)$ to $H(\tau’)$ . By [3],the homeomorphism extends to a self-homeomorphism ofpreserving $K$ .
Namely, we have an orientation-preserving homeomorphism
which$H(K,\tau)$$H(K,\tau^{l})$carriesto.
1$\tau_{1}$$\tau_{2}$$f$

$f$

$f$

$S^{3}$$\overline{f}$$\overline{f}:S^{3}arrow S^{3}$

Remark 3 In the above proof, we used the fact that every orientation-preserving
homeomorphism of a knot exterior extends to a homeomorphism of preserving
the knot, which is not always true for links. However, we may generalize arguments in this section to those for links by considering homeomorphisms of link
exteriors preserving the “peripheral structures“. We also need to assume that
one of the two compression bodies of a given Heegaard splitting of a link exterior
is a handlebody.
$S^{3}$

3. Bridge presentations of links
An n-string trivial tangle is a pair $(B, t)$ of the 3-ball and arcs properly
embedded in so that they are parallel to the boundary of . Namely, bounds
disjoint disks intogether witharcs on the boundary of . An n-bridge
sphere of a link in
is a 2-sphere which meets in $2n$ points and cuts $(S^{3}, L)$
into two n-string trivial tangles,. We call this decomposition
andof $(S^{3}, L)$ into n-string trivial tangles an n-bridge presentation (or an n-bridgedecomposition) of and denote it by. It is known that everylink admits an n-bridge presentation for some positive integer . We call a link
an n-bridge link if admits an n-bridge presentation and does not admit an
$B$

$B$

$n$

$B$

$B$

$L$

$t$

$t$

$B$

$n$

$S^{3}$$L$$(B_{1}, t_{1})$$L$$(B_{2}, t_{2})$$(B_{1}, t_{1})\cup s(B_{2}, t_{2})$

$n$

$L$

$L$

$(n-1)$ -bridge presentation.

We say that two n-bridge presentationsand(or and $S’$ in brief) of a link are equivalent (or homeomorphic) if there
exists an orientation-preserving self-homeomorphism ofwhich preserves
and sends to $S’$ . We say that and are isotopic if is isotopic to the identity
map by an isotopy preserving . Equivalence classes or isotopy classes of bridge$(B_{1}, t_{1})\cup s(B_{2}, t_{2})$$(B_{2}’, t_{2}^{l})$

$S$

$L$

$f$

$S$

$(B_{1}^{l}, t_{1}’) \bigcup_{S’}$

$S$

$L$

$S^{l}$$f$$S^{3}$$L$

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spheres of links have been studied in [9, 10, 11, 12, 19, 20, 22, 23] and so on.
Most of the results are on the uniqueness of bridge spheres of given links, andin [12] the author gave an isotopy classification of 3-bridge spheres of 3-bridge
arborescent links. (We can also obtain a homeomorphism classification of thebridge spheres from their isotopy classification.) In the paper, 3-bridge spheres
were distinguished by using the “commutator invariants” of the genus-2 Heegaard
surfaces obtained as the pre-images of the 3-bridge spheres in the double branchedcoverings ofbranched along given links. However, when becomes bigger, thecorrespondence between n-bridge spheres of links and $genus-(n-1)$ Heegaard
surfaces of 3-manifolds gets more complicated and we cannot directly apply themethods used in the case where is 3. One of the ideas which might be usefulto treat higher-index bridge presentations is using the relation between bridge
presentations of links and handlebody-links as seen in the rest of this section.
Let
be an n-bridge presentation of a link in . Note that,the closure ofis a handlebody of genus-n, which we denote by
and the closure of $(B_{2}\cup N’(L))\backslash N(L)$ is a compression-body, where $N(L)$ and$N’(L)$ are regular neighborhoods ofinsuch that $N(L)$ is contained in the$N^{l}(L)$(see Figure 4). We denote by $H(L, S;B_{1})$ the handlebody-link
interior of$N(L)\cup H_{B_{1}}$ . Another handlebody-link $H(L, S;B_{2})$ can be defined similarly, by
switching the roles ofand . In this way, we obtain two handlebody-links
from a given n-bridge presentation of a link. Then the following proposition canbe proved by an argument similar to that for Proposition 2.
$S^{3}$$n$$n$$L$$(B_{1}, t_{1})\cup s(B_{2}, t_{2})$$B_{1}\backslash N’(L)$$S^{3}$$H_{B_{1}}$$L$$S^{3}$$B_{2}$$B_{1}$

Figure 4:, suppose that two $n-bi\dot{n}dge$ presentations
Proposition 4 For a knot
inandUs’of are equivalent. Then $\{H(L, S;B_{1})$ ,$H(L, S;B_{2})\}$ and $\{H(L, S’;B_{1}’), H(L, S’;B_{2}’)\}$ are equivalent, namely,$L$$(B_{1}, t_{1})\cup s(B_{2}, t_{2})$$(B_{1}’, t_{1}’)$$S^{3}$$(B_{2}’, t_{2})$$L$(i) handlebody-links $H(L, S;B_{1})$ and $H(L, S;B_{2})$ are equivalent to $H(L, S’;B_{1}’)$and $H(L, S^{l};B_{2}’)_{f}$ respectively, $or$(ii) handlebody-links $H(L, S;B_{1})$ and $H(L, S;B_{2})$ are equivalent to $H(L, S’;B_{2}’)$and $H(L, S’;B_{1}’)$ , respectively.

As seen in Figure 4, each bridge presentation of a link gives two unknotting
tunnel systems of the link (or two Heegaard splittings of the link exterior). $Morearrow$over, we can see that an equivalence class of bridge presentations of a link gives

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the set of two equivalence classes, possibly same, of unknotting tunnel systemsof the link.

4. Towards applications
Towards applications of handlebody-links to unknotting tunnel systems or bridge
presentations of links, one can start from giving alternative proofs for some results
already known. For example, a classification of the unknotting tunnels of 2-bridge
knots (up to equivalence as in this article) is given in [15]. (We remark that theisotopy classification of the unknotting tunnels of 2-bridge knots is given in [18].
In particular, the two unknotting tunnels and of a 2-bridge knot in Figure
5 are not homeomorphic by [15]. The handlebody-links
andobtained from the two unknotting tunnels are also illustrated in the figure. Thus,ifand
are not equivalent as handlebody-links, then it implies
that and are not equivalent by Proposition 2.
$)$$\tau_{1}$$\tau_{2}$$H(K, \tau_{1})$$H(K, \tau_{1})$$\tau_{1}$$H(K, \tau_{2})$$H(K, \tau_{2})$$\tau_{2}$$r$,vsFigure 5:

Problem 5 (1) Prove that
andin Figure 5 are not equivalentas handlebody-links. How about for other unknotting tunnels of 2-bridge knots?
(2) Distinguish unknotting tunnel systems of n-bridge knots $(n>2)$ by using
the handlebody-links corresponding to them.
$H(K, \tau_{1})$$H(K, \tau_{2})$We remark that invariants which can be derived from the fundamental groupsof the complements of handlebody-links cannot distinguish the handlebody-links
obtained from unknotting tunnel systems since all such handlebody-links havecomplements with isomorphic fundamental groups, as seen in the following proposition.
Proposition 6 Let $H(K, \tau)$ be the handlebody-link obtained from a given knot
$K$ and an
unknotting tunnel system of K. Then the fundamental group of thecomplement of $H(K, \tau)$ is isomorphic to the free product of the free abelian groupof $mnk2$ and the infinite cyclic group.
$\tau$

48
in is the compression bodywhich can be decomposed by a disk into a solid torus and the productof a torus and an interval. Then the desired result can be obtained by $applying_{I}$van Kampen’s theorem.

Proof. Note that the exterior of$H(K, \tau)$$N^{l}(K\cup$$S^{3}$$\tau)\backslash N(K)$Problem 7 Distinguish n-brtdge presentations of links by comparing the sets ofhandlebody-links corresponding to them.
Recall that the unknot and a 2-bridge link admit a unique n-bridge spherefor each natural number which is bigger than or equal to 1 and 2, respectively.
(see [19, 20, 23]). Thus the simplest links admitting mutually non-equivalent(minimal) bridge spheres must be at least 3-bridge links. In fact, such examplesof 3-bridge links were given in [9, 12]. We can start with these examples to see
how useful the handlebody-links are to distinguish bridge presentations, or study
more complicated links.
$n$

We would like to thank Professor Makoto Sakuma for mentioning about
Proposition 6 which enabled us to find out mistakes in the examples given in
the talk by the author.

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$B_{7}\ddot{w}cken$$S^{3}$