## 2007 Nara Topology Seminar

1120i΁j16:20-17:50

ꏊFޗǏqwwVBSK@wZ~i[R

uҁF (HƑw)

uځFquandle ɂ錋іڕsϗʂ̍\ɂ
(On constructing knot invariants via quandles)

Abstract:
quandle Ƃ͏WƂ̏ɒꂽ񍀉Z̑gł𖞂̂łD
̏͌QɂĐωZYꋤ̐ĉƍl邱Ƃł܂D
quandle ̍\͗lXȑΏۂɑ΂ē邱Ƃł܂CɁiCӎ́j
іڂɑ΂Ă邱Ƃł܂D
̌іڂɑ΂Ēꂽ quandle ͌іڌQ͂ȕsϗʂłC
ÓTIіڂɑ΂Ă͊Ssϗʂł邱ƂmĂ܂D܂Cquandle ɂ
QƓl̎@ŃzW[^RzW[邱ƂłC̃TCN^R
TCNpČÓTIyыȖʌіڂ̕sϗʂ邱Ƃł܂D
̍uł́Cіڂ̕sϗʂ\邱ƂڕWquandle ɂĂ̏Љs
Ǝv܂DOmƂẮCʎ̌іڂȖʌіڂɂG
܂C{Iɂ͌ÓTIіڂ̒ Reidemeister moves mĂΏ\
Ǝv܂D

Abstract:
A quandle is a pair of a set and whose binary operation which satisfies some
conditions. Where the conditions is considered as properties of conjugations
of a group forgetting properties of products. We could find quandle structures
for diverse mathematical objects.
In particular, for any dimensional knot, we could define the quandle of a knot.
It is known that this quandle is stronger than the knot group of the knot and
complete invariant for classical case. On the other hand, as a group, we could
define homology / cohomology groups of a quandle. Furthermore, we could define
invariants of a classical or surface knot with cycles / cocycles of quandles.
In this talk, I will introduce quandles and how to construct invariants of a
knot via quandles. It may be sufficient to understand my talk, if you know
definitions of a classical knot and the Reidemeister moves.

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1029ij14:40-16:10

ꏊFޗǏqwwVBSK@wZ~i[R

uҁFpq(HƑw)

uځFA property of the dilatation spectrum of the chain-link
with 3 components

Abstract: Let M be a hyperbolic 3-manifold which admits surface bundle structures
over the circle. An algebraic integer, called the dilatation, is associated
to each bundle structure of M. We consider the set of dilatations associated
to all bundle structures of M, called the dilatation spectrum of M. We show
that the dilatation spectrum of $S^3 \setminus C_3$, the complement of the
chain-link with 3 components $C_3$ in the 3-sphere, contains two subsequences
such that one converges to 2 and the other converges to 1.

We also show that $S^3 \setminus C^3$ admits an n-punctured disk bundle
structure over the circle for each integer n greater than 3. This tells us
that the minimal volume among all n-punctured disk bundles over the circle
with 3 cusps is bounded above by the volume of $S^3 \setminus C_3$.

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1029ij16:20-17:50

ꏊFޗǏqwwVBSK@wZ~i[R

uҁFF(HƑw)

uځF̌vZ@ɂ
iL̋̍uɊ֘AbłDj

Abstract: Ȗʑ hyperbolic volume dilatation Ɋ֘A邢̌vZ@ƁA

̃t@Co[\eȖʑɂdilatation ƃt@Co[̃IC[
M ɂĈꕔ̃t@Co[\ɑ΂ċ̓IȌvZs܂B

ŏ dilatation Ɨ\zĂgȒAȖʑ̑
\鎖o܂BSĂA͈̑l M f[pē

ŌvZ@ĝ͕~ȂƎvĂɂeł炦悤ɁA
IȌvZ@̗pAGȎ܂łApĉ\łB

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1026ij13:00-14:00, 16F30-17F30
iԂĂ邱ƂɂӂDj

ꏊFޗǏqwwVBPK@wZ~i[T

uҁFΎ،F ({w)

uځF "Networking Seifert surgeries on knots"
(joint work with Arnaud Deruelle and Katura Miyazaki)

Abstract: How can we obtain Seifert fibered surgeries on hyperbolic knots?
In this talk, we will propose a new approach to this question.
We introduce the "Seifert Surgery Network" consisting of all the
integral Dehn surgeries on knots in the 3-sphere yielding Seifert
fiber spaces, where Seifert fiber spaces may have fibers of indices
zero as a degenerate case.
We will start with some general results and then discuss the connectivity
of the network. In particular, we will discuss which surgeries on torus
knots can be "spreaders" in the Seifert Surgery Network.

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317iyj15:00-16:00

ꏊFޗǏqwwbSK@C431-2 iwKj

uҁFProf. Sung Sook Kim(Paichai University)

uځF The Nielsen Numbers and Minimal sets of Periods
for maps on the Klein bottle

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Date: March 17(Sat.) 15:00-16:00

Room: Room: Nara Women's University C434

Speaker: Prof. Sung Sook Kim(Paichai University)

Title: The Nielsen Numbers and Minimal sets of Periods
for maps on the Klein bottle

Abstract: In this talk, we concern with the self maps on the Klein bottle.
In 1911, Bieberbach proved that any automorphism of a
crystallographic group is conjugation by an element of Aff($\R)=\R \rtimes \GL(n, \R)$. This was generalized to almost
crystallographic group. In 1995, K. B. Lee generalized this
result to all homomorphisms from isomorphisms. It can be state
as every endomorphism of flat manifolds is semi-conjugate to an
affine endomorphism. We can restate K. B. Lee's results in Klein
bottle group case as follows:

Let $\pi, \pi'\subset$ Aff($G$) be two Klein bottle groups. Then
for any homomorphism $\theta : \pi \to \pi'$, there exists
$g=(d,D) \in aff(G)= G \rx End(G)$ such that
$\theta(\alpha)\cdot g = g \cdot \alpha$ for all $\alpha \in \pi$.

Let $f: K \to K$ be any continous map on Klein bottle $K$ with the
holonomy group $\Z_2$ and let $\theta : \pi \to \pi$ be the induced
homomorphism on the fundamental group. We obtain two types of
$g=(d,D)$ by the semi-conjugate condition, and we calculate the
Nielsen numbers of periods for maps on the Klein bottle.

In terms of the Nielsen numbers of their iterates, we totally
determine the minimal sets of periods for all homotopy classes of
self maps on the Klein bottle.

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317iyj16:20-17:20

ꏊFޗǏqwwbSK@C431-2 iwKj

uҁF B(ޗǏqw)

uځF Distances of knots and Morimoto's Conjecture on the super
additive phenomena of tunnel numbers of knots

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Date: March 17(Sat.) 16:20-17:20

Room: Nara Women's University C434

Speaker: Tsuyoshi Kobayashi(Nara Women's Univ.)

Title: Distances of knots and Morimoto's Conjecture on the super
additive phenomena of tunnel numbers of knots

Abstract: Let $K_i$ ($i=1,2$) be knots in the 3-sphere $S^3$, and
$K_1 \# K_2$ their connected sum. We use the notation $t(\cdot)$
to denote tunnel number of a knot. It is well known that the
following inequality holds in general.

$$t(K_1 \# K_2) \leq t(K_1) + t(K_2) +1.$$

We say that a knot $K$ in a closed orientable manifold $M$ admits a
$(g,n)$ position if there exists a genus $g$ Heegaard surface
$\sigma \subset M$, separating $M$ into the handlebodies
$H_1$ and $H_2$, so that $H_i \cap K$ ($i=1,2$) consists of $n$
arcs that are simultaneously parallel into $\partial H_i$. It is
known that if $K_i$ ($i=1$ or 2) admits a $(t(K_i),1)$ position
then equality does not hold in the above. Morimoto proved that if
$K_1$ and $K_2$ are m-small knots then the converse holds, and
conjectured that this is true in general (K.Morimoto, Math. Ann.,
317(3):489--508, 2000).

Morimoto's Conjecture
Given knots $K_1,\ K_2 \subset S^3$, $t(E(K_1 \# K_2)) < t(E(K_1)) + t(E(K_2))+1$ if and only if for $i=1$ or $i=2$, $K_i$ admits a
$(t(K_i),1)$ position.

In this talk, we describe how to show the existence of conterexamples
to this conjecture by making use of the \lq distance\rq of knots.

221ij16:30-17:30

ꏊFޗǏqwwbSK@C431 iwKj

uҁFq玁(ʑw)

uځFDNA and lens space surgery

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Date: February 21 (Wed.) 16:30-17:30

Room: Nara Women's University C431

Speaker: Prof. Koya Shimokawa(Saitama University)

Title: DNA and lens space surgery

Abstract: In this survey talk I will show how lens space surgeries on knots
in the 3-sphere can be applied to the study of enzymes acting on DNA.
In particular, I will discuss how our result (joint with Hirasawa)
on lens space surgeries on strongly invertible knots has been put
to use by biologists.

wEޗǏqwg|W[Z~i[

129ij15:30-17:00

ꏊFޗǏqwwbSK@C431 iwKj

uҁFProf. Jonathan Hillman(Vhj[w)

uځFFiniteness conditions, Novikov rings and Mapping Tori

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Date: January 29 (Mon) 15:30-17:00

Room: Nara Women's University C431

Speaker: Prof. Jonathan Hillman(University of Sydney)

Title: Finiteness conditions, Novikov rings and Mapping Tori

Abstract: In 1962 Stallings gave a simple and natural algebraiccriterion for
a 3-manifold to be a mapping torus, in other words, to fibre over
the circle. The corresponding infinite cyclic covering space is then
homotopy equivalent to a closed surface, and so is a $PD_2$-complex.
An analogous result for infinite cyclic coverings of higher dimension
manifolds with fundamental group $Z$ was found by Milnor, and Quinn
and Gottlieb extended this to more general fibrations of $PD$-complexes
over $PD$-complexes.
We are interested in simplifying the hypotheses of such homotopy
fibration theorems. Our main result is the following:
Theorem. Let $p:M'\to M$ be an infinite cyclic coverof a closed
$n$-manifold (with $n\not=4$). Then $M'$ is a $PD_{n-1}$-complex if
and only if $\chi(M)=0$ and $M'$ has finite $[(n-1)/2]$-skeleton (up to homotopy).
The theorem remains true for $n=4$ if we replace $PD_3$-complex" by
a slightly weaker notion.
This result may be regarded as lying between those of Milnor and
Gottlieb-Quinn. Our arguments are purely homological, and take full
advantage of Poincare duality in $M$. Examples deriving from high
dimensional simple knots show that the dimension bounds are best
possible.
This is joint work with D.H.Kochloukova, of Brazil. I have not met her;
our collaboration is a happy consequence of the travels of Peter Zvengrowski.

wEޗǏqwg|W[Z~i[

18ij15:30-17:00i̓͏jłj

ꏊFޗǏqwwbSK@C431 iwKj

uҁFProf. Craig Hodgson({w)

uځFIntroduction to hyperbolic 3-manifolds

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Date: January 8 (Mon) 15:30-17:00

Room: Nara Women's University C431

Speaker: Prof. Craig Hodgson(The University of Melbourne)

Title: Introduction to hyperbolic 3-manifolds

Abstract: This talk will give an introduction to hyperbolic geometry, describe
examples of hyperbolic 3-manifolds, and survey some of the main
results on the classification of hyperbolic 3-manifolds. The talk is
aimed at a general audience, and should be suitable for beginning