数理解析研究所講究録
第 1777 巻 2012 年 107-109

107

PROBLEM SESSIONEDITED BY TERUAKI KITANO

1. HIROSHI GODA
Let$L$

be a link in$\tau\cap L=\partial\tau$$S^{3}$.

and$\tau=t_{1}\cup\cdots Ut_{n}$arcs embedded inDeflnition 1.1. is an unknotting tunnel system of if
genus $(n+1)$ handlebody.
$L$$\tau$$S^{3}$

such that$S^{3}\backslash N^{o}(L\cup\tau)$

is aWe define the tunnel number of by$t(L)$ $:= \min$ {: unknotting tunnel system of }When $t(L)=1,$ $\tau(=t_{1})$ is called an unknotting tunnel. Let denote aminimal bridge sphere of a knot or a link .
$L$$n|\exists t_{1}U\cdots Ut_{n}$

$L$

$S$

$K$

$L$

Theorem 1.2 (Goda-Scharlemann-Thompson). If is a knot withthen the unknotting tunnel may be isotoped into .
$K$$t(\mathscr{K}=1$,$S$$\tau$The next theorem is a key to prove Theorem 1.2.

Theorem 1.3 (Gordon-Reid). If $t(K)=1$ , thentangle decomposition.

$K$

does not have an essentialConjecture 1.4 (Scharlemann). If does not have an essential tangle decomposition, then all arcs in the unknotting tunnel system may be isotoped
into .
$K$$\tau$$S$(1) Prove this conjecture.
Problem 1.5.
(2) Find a knot
with an essential tangle decomposition and its unknotting tunnet system such that cannot be isotoped to the minimal bridge sphere .
(3) Consider the link case.
$K$$\tau$$\tau$$S$

REFERENCES
1. Goda, Hiroshi; Scharlemann, Martin; Thompson, Abigail, Levelling an unknomng
tunnel, Geom. Topol. 4 (2000), 243-275.
2. Gordon, C. McA.; Reid, A. W., Tangle decompositions of tunnel number one knots
and links, J. Knot Theory Ramifications 4 (1995), no. 3, 389-409.

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EDITED BY TERUAKI KITANO

2. TAKAYUKI MORIFUJI
Recently Friedl and Vidussi showed in [3] that the twisted Alexanderpolynomials corresponding to all finite representations detect fibered 3-manifolds.
Let K be a knot inand $G(K)$ its knot group. We consider the twistedof a hyperbolic knot K associated with aAlexander polynomial
discrete faithful representation : $G(K)arrow SL(2,\mathbb{C})$ .
$S^{3}$$\Delta_{Kp0}(t)$$\rho_{0}$Conjecture 2.1 (Dunfield-Friedl-Jackson [1]). The twistedAlexander polynomial
detects the genus ofdetects fibered knots. Moreover
K. That is, $\deg\Delta_{K_{\theta 0}}(t)=4g(K)-2$ holds.
$\Delta_{K,\rho_{0}}(t)$$\Delta_{K_{\theta 0}}(t)$Remark 2.2. For any representation$\rho$: $G(K)arrow SL(2, \mathbb{C})$, it is known that

holds. Dunfield-Friedl-Jackson checked in [1] thatthe conjecture is true for hyperbolic knots up to 15-crossings.

$\deg\Delta_{Kp}(t)\leq 4g(K)-2$

Question 2.3 (Friedl [2]). Is there an example of a hyperbolic knot
with a $faithfi\iota l$ representation :such that $\deg\Delta_{Kp}(t)\neq$$4g(K)-2$ ?

$K\subset S^{3}$$\rho$$G(\mathscr{T}arrow SL(2,\mathbb{C})$

Remark 2.4 (Friedl [2]). There is a hyperbolic knot with a faithful andirreducible representation : $G(K)arrow SL(3,\mathbb{C})$ such that
does notdetect the genus.
$K$$\Delta_{Kp}(t)$$\rho$REFERENCES
1. N. Dunfield, S. Friedl and N. Jackson, Twisted Alexander polynomials of hyperbolic
knots, arXiv: 1108.3045.
2. S. Friedl, Private communication, 2011.
3. S. Friedl and S. Vidussi, Twisted Alexander polynomials detect fibered 3-mmfolds,
Ann. of Math. (2) 173 (2011), 1587-1643.

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PROBLEM SESSION

3. MASAAKI SUZUKILet K be a knot and $G(K)$ its knot group. Let
be the twisted
Alexander polynomial associated to a representation of $G(K)$ .
$\Delta_{K,\rho}(t)$$\rho$

Theorem 3.1 (Kitano-Suzuki-Wada). Let
be knots. If there exist aprime number p and a representation : $G(K_{2})arrow SL(2;Z/pZ)$ such thatis not divisible byfor any representation : $G(K_{1})arrow$$SL(2;Z/pZ)$, then there does not exist an epimorphism $G(K_{1})arrow G(K_{2})$ .
Problem 3.2. Is the converse is true ?
$K_{1},$ $K_{2}$$\rho_{2}$$\Delta_{K_{1}p_{1}}(t)$$\Delta_{K_{2}p_{2}}(t)$$\rho_{1}$4. TAKUYA SAKASAI
Let be a free group of rank and be a compact Lie group. Consider
a nomal subgroup, for example, $[[F, F], F],$ $[[F,F], [F, F]]$ , etc.
$F$

$n$

$G$

$\Gamma\triangleleft F$Problem 4.1. Define a good measure in the space$R_{\Gamma}(G)$$:=Hom(F/\Gamma,G)$.

The background of this problem is as follows. Twisted invariants such as
twisted Alexander polynomial,
twisted Reidemeister torsion,
Atiyah-Patodi-Singer’s p-invariant,
are basically invariants of a manifold (or link) together with a representation of some groupassociated with . Fixing the manifold , we may
regard these invariants as functions on the representation space.
To get an invariant of itself, we would like to consider the “totality” of such a function, which often behaves well on. For example, Levine [1] defined G-concordance invariants of links after observing
that Atiyah $-Patodiarrow$ Singer’s -invariant gives bounded, continuous and homology cobordism invariant functions on representation spaces away fromsome singular loci.
To answer the above (somewhat abstract) problem derives us to define invariants of manifolds by integrations of twisted invariants on representation$\bullet$$\bullet$$\bullet$

$X$

$X$

$F/\Gamma$$X$$R_{\Gamma}(G)$$X$$R_{\Gamma}(G)$$\rho$

spaces.

REFERENCES[1] J. Levine, Link invariants via the eta invariant, Comment. Math. Helv. 69 (1994),

82-119.