Topological invariants in low dimensional topology
last update: 2017.11.6
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| Group photo |
This workshop was supported by
JSPS KAKENHI Grant Numbers JP16K05161 and JP17K05252.
Dates: 31,October(Tue.)(2 p.m.) - 2,November(Thu.)(around 5 p.m.), 2017
Venue:Day 1(31th Oct.):Midsize Conference Room (4th floor of Matsue Terrsa
map )
Day 2-3(1-2.Nov.):Multipurpose Hall (2nd floor of Interdisciplinary Faculty of Science and Engineering Building III,
Shimane univ.map)
Program (PDF version is here）
31th Oct.(Tue.)
14:00-15:00 Tadayuki Watanabe (Shimane univ.) abstract
An equivariant perturbative invariant of 3-manifolds with $b_1=1$
15:15-16:15 Tatsuro Shimizu (Kyoto univ.) abstract
On the Bott-Cattaneo's Chern-Simons perturbation theory
16:30-17:30 Andrew Kricker (Nanyang Technological univ.) abstract
What is the l^2-Alexander invariant and how can we approximate it?
1st Nov.(Wed.)
9:30-10:30 Sakie Suzuki (Kyoto univ.) abstract
The universal quantum invariant and colored ideal triangulations
11:00-12:00 Christine Lescop (Univ. Grenoble Alpes) abstract
On the cube of the Blanchfield pairing and on its generalization as a universal equivariant finite type knot invariant
2nd Nov.(Thur.)
9:30-10:30 Delphine Moussard (Kyoto univ.) abstract
A-ribbon 2-knots and factorized Alexander polynomial
11:00-12:00 Atsushi Mochizuki (Kyoto univ.) abstract
On the Casson-Walker invariant of 3-manifolds admitting genus one open book decompositions
14:00-15:00 Yuta Nozaki (Univ. of Tokyo) abstract
Homology cobordisms over a surface of genus one
15:30-16:30 Masatoshi Sato (Tokyo Denki univ.) abstract
On the homology group of the genus three handlebody mapping class group
(last update:2017.10.12.)
Abstracts (PDF version is here)
Tadayuki Watanabe
Title: An equivariant perturbative invariant of 3-manifolds with $b_1=1$
Abstract: In this talk, I will explain that a topological invariant for 3-manifold
$M$ with $b_1(M)=1$ can be constructed by applying Fukaya's Morse homotopy theoretic
approach for Chern--Simons perturbation theory to a local coefficient system on $M$
of rational functions associated to the maximal free abelian covering of $M$. The
invariant takes values in Garoufalidis--Rozansky's space of Jacobi diagrams whose
edges are colored by rational functions. It is expected that the invariant gives a
lot of nontrivial finite type invariants of 3-manifolds. The definition of the
invariant was strongly inspired by (and partially equivalent to) the equivariant
invariant of Lescop.
Tatsuro Shimizu
Title: On the Bott-Cattaneo's Chern-Simons perturbation theory
Abstract: The Chern-Simons perturbation theory was established by Kontsevich, Axelrod and Singer in 1991.
This gives an invariant of 3-manifold with a local system on the given manifold.
Bott and Cattaneo gave a purely topological construction of Chern-Simons perturbation theory.
In this talk we revisit their construction and define a kind of obstruction class.
If this obstrcution class does not vanish, the original construction of Bott-Cattaneo should be modified.
We show the modified construction. This construction is joint work with Alberto Cattaneo.
Andrew Kricker
Title: What is the l^2-Alexander invariant and how can we approximate it?
Abstract:
The classical Alexander polynomial marks the beginning of mathematical knot theory.
The l^2-Alexander invariant is a recent variation due to Li-Zhang (2006) and Dubois-Friedl-Luck (2014) where you twist the classical construction by the infinite-dimensional representation consisting of square-summable series of elements of the fundamental group.
This invariant has remarkable properties - such as the fact that it determines the volume of a hyperbolic knot, determines the Thurston norm, and detects infinitely many knots.
In this talk I'll introduce this invariant and survey what is known and unknown about it.
It is a very strong invariant, but also is almost impossible to calculate. I'll also talk about some ongoing projects investigating its approximation.
Sakie Suzuki
Title: The universal quantum invariant and colored ideal triangulations
Abstract: The Drinfeld double of a finite dimensional Hopf algebra is a
quasi-triangular Hopf algebra with the canonical element as the universal
R-matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon
element. The universal quantum invariant of framed links is constructed
using a ribbon Hopf algebra. In that construction, a copy of the
universal R-matrix is attached to each crossing, and invariance under the
Reidemeister III move is shown by the quantum Yang-Baxter equation of the
universal R-matrix.
On the other hand, the Heisenberg double of a finite dimensional Hopf
algebra has the canonical element (the S-tensor) satisfying the pentagon
relation. In this talk we reconstruct the universal quantum invariant using
the Heisenberg double, and extend it to an invariant of equivalence
classes of colored ideal triangulations of 3-manifolds up to colored
moves. In this construction, a copy of the S-tensor is attached to each
tetrahedron, and invariance under the colored Pachner (2,3) moves is shown
by the pentagon relation of the S-tensor.
Christine Lescop
Title: On the cube of the Blanchfield pairing and on its generalization as a universal equivariant finite type knot invariant
Abstract:
In 1999, Greg Kuperberg and Dylan Thurston proved that the Casson
invariant of an integer homology $3$-sphere $M$ can be seen as the
integral
of the cube of a differential $2$-form, which represents the linking
form in a two-point configuration space of the manifold $M$.
We will define an invariant of knots in such manifolds as a similar
"cube" of the equivariant linking form, which is defined on the
infinite cyclic cover of the knot complement.
The obtained invariant will be defined as an equivariant algebraic
intersection number in an equivariant configuration space.
It is the degree one part of a more general equivariant invariant
$Z_e$, which shares many properties -including a fundamental
universality property- with the Kricker lift of the Kontsevich integral.
This Kricker lift contains all the Vassiliev knot invariants and it
provides the space of these finite type invariants with a strong
additional structure.
We will introduce $Z_e$ and discuss its main properties.
Delphine Moussard
Title: A-ribbon 2-knots and factorized Alexander polynomial
Abstract: For classical knots, the Alexander polynomial is known to be symmetric and
to have a good factorization property on ribbon knots. In contrast, any Laurent
polynomial with integer coefficients, equal to 1 when evaluated at 1, can be
realized as the Alexander polynomial of a ribbon 2-knot. We give a topological
condition on 2-knots which implies the factorization of the Alexander polynomial. To
this end, we introduce an alternative notion of ribbon 2-knots, the A-ribbon
2-knots, which are 2-knots that bound an immersed 3-ball with ribbon singularities
homeomorphic to annuli.
This is a joint work with Emmanuel Wagner.
Atsushi Mochizuki
Title: On the Casson-Walker invariant of 3-manifolds admitting genus one open book decompositions
Abstract: In this talk, I compute the Casson-Walker invariant of genus one
open books in two ways; one is based on the combinatorial calculation of
the degree one part of the LMO invariant, and the other is based on the
action of the LMO invariant of mapping cylinders on the space of Jacobi
diagrams on two intervals. Especially, in the latter case, I construct a
representation of a central extension of the mapping class group of the
genus one compact surface with one boundary.
Yuta Nozaki
Ttle: Homology cobordisms over a surface of genus one
Abstract: Morimoto proved that some lens spaces do not admit any open book
decomposition whose page is a surface $\Sigma_{1,1}$.
In contrast, we show that every lens space is obtained as the closure of
a homology cobordism over $\Sigma_{1,1}$.
The proof is based on the Chebotarev density theorem and binary
quadratic forms in number theory.
Masatoshi Sato
Title: On the homology group of the genus three handlebody mapping class group
Abstract: We compute low-dimensional rational homology of the genus three handlebody
mapping class group.
The handlebody mapping class group acts on a simplicial complex called the
meridian disk complex, and McCullough showed that it is contractible.
The stabilizer of a vertex is the mapping class group of a genus two
handlebody with two marked disks, and the stabilizer of one of edges are the
mapping class groups of a solid torus with four marked disks.
In this talk, we explain a method to compute the homology groups of these
stabilizers and the genus three handlebody mapping class group.
Speakers:
Andrew Kricker (Nanyang Technological univ.)
Christine Lescop (Univ. Grenoble Alpes)
Atsushi Mochizuki (Kyoto univ.)
Delphine Moussard (Kyoto univ.)
Yuta Nozaki (Univ. of Tokyo)
Masatoshi Sato (Tokyo Denki univ.)
Tatsuro Shimizu (Kyoto univ.)
Sakie Suzuki (Kyoto univ.)
Tadayuki Watanabe (Shimane univ.)
Organizers:
Teruaki Kitano (Soka univ.)
Tatsuro Shimizu (Kyoto univ.)
Tadayuki Watanabe (Shimane univ.)