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Tsuyoshi KOBAYASHI Home Page

2007 Nara Topology Seminar

11寧20擔乮壩乯16:20-17:50

応強丗撧椙彈巕戝妛棟妛晹怴B搹係奒丂悢妛僙儈僫乕幒俁

島墘幰丗堜忋 曕巵 (搶嫗岺嬈戝妛)

島墘戣栚丗quandle 偵傛傞寢傃栚晄曄検偺峔惉偵偮偄偰
(On constructing knot invariants via quandles)

Abstract:
quandle 偲偼廤崌偲偦偺忋偵掕媊偝傟偨擇崁墘嶼偺慻偱偁傞忦審傪枮偨偡傕偺偱偡丏
偙偺忦審偼孮偵偍偄偰愊墘嶼傪朰傟嫟栶偺惈幙傪巆偟偨傕偺偲峫偊傞偙偲偑偱偒傑偡丏
quandle 偺峔憿偼條乆側懳徾偵懳偟偰摫擖偡傞偙偲偑偱偒傑偡偑丆摿偵乮擟堄師尦偺乯
寢傃栚偵懳偟偰傕掕媊偡傞偙偲偑偱偒傑偡丏
偙偺寢傃栚偵懳偟偰掕媊偝傟偨 quandle 偼寢傃栚孮傛傝傕嫮椡側晄曄検偱偁傝丆
屆揟揑寢傃栚偵懳偟偰偼姰慡晄曄検偱偁傞偙偲偑抦傜傟偰偄傑偡丏傑偨丆quandle 偵偼
孮偲摨條偺庤朄偱儂儌儘僕乕乛僐儂儌儘僕乕傪掕媊偡傞偙偲偑偱偒丆偦偺僒僀僋儖乛僐
僒僀僋儖傪梡偄偰屆揟揑媦傃嬋柺寢傃栚偺晄曄検傪掕媊偡傞偙偲偑偱偒傑偡丏
偙偺島墘偱偼丆寢傃栚偺晄曄検傪峔惉偡傞偙偲傪栚昗偵quandle 偵偮偄偰偺徯夘傪峴偄
偨偄偲巚偄傑偡丏慜採抦幆偲偟偰偼丆彮偟偩偗堦斒師尦偺寢傃栚傗嬋柺寢傃栚偵傕怗傟
傑偡偑丆婎杮揑偵偼屆揟揑寢傃栚偺掕媊偲 Reidemeister moves 傪抦偭偰偄傟偽廫暘偩
偲巚偄傑偡丏

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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10寧29擔乮寧乯14:40-16:10

応強丗撧椙彈巕戝妛棟妛晹怴B搹係奒丂悢妛僙儈僫乕幒俁

島墘幰丗嬥塸巕巵(搶嫗岺嬈戝妛)

島墘戣栚丗A 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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10寧29擔乮寧乯16:20-17:50

応強丗撧椙彈巕戝妛棟妛晹怴B搹係奒丂悢妛僙儈僫乕幒俁

島墘幰丗崅戲岝旻巵(搶嫗岺嬈戝妛)

島墘戣栚丗偄偔偮偐偺寁嶼婡幚尡偵偮偄偰
乮忋婰偺嬥巵偺島墘偵娭楢偟偨偍榖偱偡丏乯

Abstract: 嬋柺懇偺 hyperbolic volume 偲 dilatation 偵娭楢偡傞偄偔偮偐偺寁嶼婡幚尡偲丄
偦偙偐傜摼傜傟偨娤嶡傪徯夘偟傑偡丅摿偵丄埲壓偺俀偮偵廳揰傪偍偔偮傕傝偱偡丅

柍尷屄偺僼傽僀僶乕峔憿傪嫋梕偡傞嬋柺懇偵偍偗傞dilatation 偲僼傽僀僶乕偺僆僀儔乕
悢偵偮偄偰偼堦掕偺娭學偑偁傞帠偑棟榑揑偵傢偐偭偰偄傑偡丅俁惉暘偺 chain link 偺
曗嬻娫 M 偵偍偄偰堦晹偺僼傽僀僶乕峔憿偵懳偟偰嬶懱揑側寁嶼傪峴偄傑偟偨丅

嵟彫偺 dilatation 傪帩偮偲梊憐偝傟偰偄傞慻昍偺懓偑偁傝丄偦偙偐傜嬋柺懇偺懓
傪峔惉偡傞帠偑弌棃傑偡丅偦傟傜慡偰偑丄幚偼堦偮偺懡條懱 M 傪僨乕儞庤弍偟偰摼傜
傟傞偙偲偑傢偐傝傑偟偨丅

帺暘偱寁嶼婡傪巊偆偺偼晘嫃偑崅偄側偀偲巚偭偰偄傞曽偵傕恊偟傫偱傕傜偊傞傛偆偵丄
弶曕揑側寁嶼婡偺棙梡偐傜丄暋嶨側幚尡傑偱傪丄幚椺傪梡偄偰夝愢偡傞梊掕偱偡丅

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10寧26擔乮嬥乯13:00-14:00, 16丗30-17丗30
 乮帪娫偑暘偐傟偰偄傞偙偲偵偛拲堄偔偩偝偄丏乯

応強丗撧椙彈巕戝妛棟妛晹怴B搹侾奒丂悢妛僙儈僫乕幒俆

島墘幰丗栁庤栘岞旻巵 (擔杮戝妛)

島墘戣栚丗 "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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3寧17擔乮搚乯15:00-16:00

応強丗撧椙彈巕戝妛棟妛晹俠搹係奒丂C431-2 乮悢妛墘廗幒乯

島墘幰丗Prof. Sung Sook Kim(Paichai University)

島墘戣栚丗 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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3寧17擔乮搚乯16:20-17:20

応強丗撧椙彈巕戝妛棟妛晹俠搹係奒丂C431-2 乮悢妛墘廗幒乯

島墘幰丗彫椦 婤(撧椙彈巕戝妛)

島墘戣栚丗 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.

2寧21擔乮悈乯16:30-17:30

応強丗撧椙彈巕戝妛棟妛晹俠搹係奒丂C431 乮悢妛墘廗幒乯

島墘幰丗壓愳峲栫巵(嶉嬍戝妛)

島墘戣栚丗DNA 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.

戝嶃戝妛丒撧椙彈巕戝妛崌摨僩億儘僕乕僙儈僫乕

1寧29擔乮寧乯15:30-17:00

応強丗撧椙彈巕戝妛棟妛晹俠搹係奒丂C431 乮悢妛墘廗幒乯

島墘幰丗Prof. Jonathan Hillman(僔僪僯乕戝妛)

島墘戣栚丗Finiteness 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.

戝嶃戝妛丒撧椙彈巕戝妛崌摨僩億儘僕乕僙儈僫乕

1寧8擔乮寧乯15:30-17:00乮偙偺擔偼廽擔偱偡乯

応強丗撧椙彈巕戝妛棟妛晹俠搹係奒丂C431 乮悢妛墘廗幒乯

島墘幰丗Prof. Craig Hodgson(儊儖儃儖儞戝妛)

島墘戣栚丗Introduction 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
graduate students.