Low dimensional topology and number theory
March 17-20, 2009
Soft Research Park Center, Fukuoka
ProgramMarch 17
11:00 – 12:00
Susan Williams (University of South Alabama)
Twisted Alexander invariants and two-bridge knots
13:30 – 14:30
Yuichi Kabaya (Tokyo Institute of Technology)
A method for ﬁnding ideal points from an ideal triangulation and its application
14:50 – 15:50
Stavros Garoufalidis (Georgia Institute of Technology)
Asymptotics of classical spin networks
March 18
10:00 – 11:00
Daniel Silver (University of South Alabama)
On a theorem of Burde and de Rham
11:20 – 12:20
Takahiro Kitayama (The University of Tokyo)
Torsion volume forms and twisted Alexander functions on character varietiesof knots
14:00 – 15:00
Kazuo Habiro (RIMS, Kyoto University)
On the Witten-Reshetikhin-Turaev invariant and analytic functions on rootsof unity1

15:20 – 16:20
Don Zagier (Max Planck Institute)
Modular properties of topological invariants and other q-series
March 19
10:00 – 11:00
Jonathan Hillman (University of Sydney)
Indecomposable P D3 -complexes
11:20 – 12:20
Baptiste Morin (University of Bordeaux)
On the Weil-étale fundamental group of a number ﬁeld
14:00 – 15:00
Ken-ichi Sugiyama (Chiba University)
On a geometric analog of Iwasawa conjecture
15:00 – 15:25
Walter Neumann (Columbia University)
Universal abelian covers in geometry and number theory
March 20
10:00 – 11:00
Thang Le (Georgia Institute of Technology)
Hyperbolic volumes, Mahler measure and homology growth
11:20 – 12:20
Shinya Harada (Kyushu University)
Hasse-Weil zeta function of absolutely irreducible SL2 -representations of theﬁgure 8 knot group
14:00 – 15:00
Jinsung Park (Korea Institute for Advanced Study)
Eta invariant and Selberg zeta function of odd type for convex co-compact
hyperbolic manifolds2

15:20 – 16:20
Tomoyoshi Yoshida (Tokyo Institute of Technology)
An asymptotic behavior of basis elements of conformal blocks of SU(2) WZWmodel in the large level3

Low dimensional topology and number theory II
March 15-18, 2010
The University of Tokyo, Tokyo
ProgramMarch 15
9:30 – 10:30
Eriko Hironaka (Florida State University)
Mapping classes with small dilatation
11:00 – 12:00
Jonathan Hillman (University of Sydney)
Embedding 3-manifolds with circle actions
13:20 – 14:20
Daniel Silver (University of South Alabama)
Twisted Alexander invariants
14:40 – 15:40
Takashi Hara (The University of Tokyo)
Reidemeister torsion, p-adic zeta function and its non-abelianization
16:00 – 17:00
Yoshikazu Yamaguchi (The University of Tokyo)
On the twisted Alexander polynomial for cyclic covers over knot exteriors
March 16
9:30 – 10:30
Yoshiyuki Yokota (Tokyo Metropolitan University)
On the complex volumes of hyperbolic knots
11:00 – 12:00
Tudor Dimofte (California Institute of Technology)
TQFT and the Volume Conjecture4

13:20 – 14:20
Hiroyuki Fuji (Nagoya University)
The Volume Conjecture and Topological Recursion
14:40 – 15:40
Toshie Takata (Niigata University)
On the SO(N ) and Sp(N ) free energy of a closed oriented 3-manifold
16:00 – 17:00
Susama Agarwala (California Institute of Technology)
The β-function over curved space-time under ζ-function regularization
18:00
Banquet (UNOSATO)
March 17
9:30 – 10:30
Thang Le (Georgia Institute of Technology)
Integrality of the Witten-Reshetikhin-Turaev invariant of 3-manifolds anduniﬁed invariant for rational homology 3-spheres
11:00 – 12:00
Kazuo Habiro (RIMS, Kyoto University)
On the reduced colored Jones polynomials of links
13:20 – 14:20
Javier López Peña (Queen Mary University of London)
Cyclotomy and the ﬁeld with one element
14:40 – 15:40
Sachiko Ohtani (National Defense Academy)
An Analogy between Deformation Spaces of Representations of Knot Groups
and Galois Groups
16:00 – 17:00
Baptiste Morin (California Institute of Technology)
5

The Weil-étale topos and Deninger’s dynamical system
March 18
9:30 – 10:30
Ken-ichi Sugiyama (Chiba University)
On an analogy between number theory and hyperbolic geometry
11:00 – 12:00
Takahiro Kitayama (The University of Tokyo)
Non-commutative Reidemeister torsion and Morse-Novikov theory
13:20 – 14:20
Takayuki Morifuji (Tokyo University of Agriculture and Technology)
Twisted Alexander polynomials and nonabelian representations of 2-bridge
knot groups
14:40 – 15:40
Susan Williams (University of South Alabama)
Alexander-Lin twisted polynomials

6

Low dimensional topology and number theory III
March 14-17, 2011
Nishijin Plaza, Fukuoka
ProgramMarch 14
9:30 – 10:30
Daniel Silver (University of South Alabama)
Twisted Alexander polynomials of twisted knots
10:50 – 11:50
Takahiro Kitayama (The University of Tokyo)
Non-commutative Reidemeister torsion for homology cylinders
13:30 – 14:30
Susan Williams (University of South Alabama)
Bounding the degree and roots of twisted Alexander polynomials
14:50 – 15:50
Jean Raimbault (Institut de Jussieu)
Asymptotic behaviour of homology
16:10 – 17:10
Thang Le (Georgia Institute of Technology)
Integrality of the Wittten-Reshetikhin-Turaev SU(2) invariants
March 15
9:30 – 10:30
Adam Sikora (SUNY, Buﬀalo)
Idele class ﬁeld theory for low dimensional topology
10:50 – 11:50
Norihiko Minami (Nagoya Institute of Technology)
On the Connes-Consani-Soulé type zeta function for F1 -schemes713:30 – 14:30
Kazuhiro Hikami (Naruto University of Education)
Superconformal algebra, mock theta function, and Mathieu moonshine
14:50 – 15:50
Anton Mellit (University of Cologne)
Cell decompositions of Siegel modular spaces and Morse theory
16:10 – 17:10
Shinya Harada (Korea Institute for Advanced Study)
Hasse-Weil zeta functions of SL2 -character varieties of some arithmetic hyperbolic 3-manifolds
18:00
BanquetMarch 16
9:30 – 10:30
Kathleen Petersen (Florida State University)
Character Varieties of Some Families of 3-manifolds
10:50 – 11:50
Akishi Kato (The University of Tokyo)
Geometry of colored Jones polynomials
13:30 – 14:30
Kazuo Habiro (RIMS, Kyoto University)
Quantum fundamental groups and quantum representation varieties for 3manifolds
14:50 – 15:50
Hidekazu Furusho (Nagoya University)
On Associators
16:10 – 17:10
Akihiro Tsuchiya (IPMU, The University of Tokyo)
8

Conformal Field Theory and Quantum Group
March 17
9:30 – 10:30
Stéphan Baseilhac (Université Montpellier 2)
Asymptotic rigidity of the quantum hyperbolic invariants
10:50 – 11:50
Jinseok Cho (Waseda University)
Optimistic limit of the colored Jones invariant
13:30 – 14:30
Hiroyuki Fuji (Nagoya University)
Surface Operator and Topological String
14:50 – 15:50
Takeshi Torii (Okayama University)
On the K(n)-local category9Low dimensional topology and number theory IV
March 12-15, 2012
Kyushu University, Fukuoka
ProgramMarch 1210:00 – 10:50
Makoto Ozawa (Komazawa University)
On the Neuwirth conjecture for knots (Joint work with J. Hyam Rubinstein)
11:00 – 11:50
Shinya Harada (Korea Institute for Advanced Study)
SL2 character varieties and zeta functions of arithmetic two-bridge links
14:00 – 14:50
Toshitake Kohno (The University of Tokyo)
Quantum and homological representations of braid groups
15:00 – 15:25
Fumiya Amano (Kyushu University)
On Milnor primes
15:35 – 16:25
Hiroaki Nakamura (Okayama University)
On some arithmetic monodromy invariant for once punctured torus
March 13
10:00 – 10:50
Kentaro Ihara (Max Planck Institute)
Iterated integral of modular forms
11:00 – 11:50
Thang Le (Georgia Institute of Technology)
On the stability of the colored Jones polynomial1014:00 – 14:50
Kazuo Habiro (RIMS, Kyoto University)
To be announced
15:00 – 15:50
Marc Culler (University of Illinois at Chicago)
Peripherally elliptic representations and character varieties
18:00
Banquet (Tamaru in Momochi)
March 14
10:00 – 10:50
Kazuhiro Ichihara (Nihon University)
Exceptional surgeries on Montesinos knots (Joint work with In Dae Jong)
11:00 – 11:50
Yoriko Kodani (Nara Women’s University)
A sequence of new bridge indices for links each of which has a trivial knot
component
14:00 – 14:50
Makoto Sakuma (Hiroshima University)
On Heckoid groups associated with 2-bridge links
15:00 – 15:25
Jun Ueki (Kyushu University)
On Iwasawa invariants for 3-manifolds
15:35 – 16:25
Joan Porti (Universitat Autonoma de Barcelona)
Reidemeister torsion and volume for hyperbolic knots

11March 15
10:00 – 10:50
Hidekazu Furusho (Nagoya University)
On Galois action on knots
11:00 – 11:50
Tudor Dimofte (IAS, Princeton)
Nonabelian Torsion and the Neumann-Zagier Equations

12

Low dimensional topology and number theory V
March 11-14, 2013
Soft Research Park Center, Fukuoka
ProgramMarch 1110:00 〜 11:00
Yasushi Mizusawa (Nagoya Institute of Technology)
Iwasawa invariants and Greenberg type problem for links(a joint work with Teruhisa Kadokami)
11:10 〜 12:00
Jun Ueki (Kyushu University)
On Iwasawa invariants for links and ”Kida’s formula”
14:00 〜 15:00
Kathleen Petersen (Florida State University)
Character Varieties and Dehn Filling
15:10 〜 16:10
Takahiro Kitayama (The University of Tokyo)
On an analogue of Culler-Shalen theory for higher-dimensional representations
17:30 Bunquet (Italian at La Manina)
March 1210:00 〜 11:00
Ayako Ido (Nara Women’s University)
Heegaard splitting with distance exactly n(joint work with Yeonhee Jang and Tsuyoshi Kobayashi)
11:10 〜 12:10
Steve Boyer (Université du Quebec à Montréal)13

L-spaces, left-orderability and foliations
14:00 〜 15:00
Atsushi Katsuda (Kyushu University)
Toward a geometric density theorem for nilpotent extensions
15:10 〜 16:10
Jinsung Park (Korea Institute for Advanced Study)
Bergman tau function and Chern-Simons invariant
March 13
10:00 〜 11:00
Shunji Moriya (Kyoto University)
Homology of space of long knots and operad formality
11:10 〜 12:10
Kazuo Habiro (RIMS, Kyoto University)
Kirby calculus for null-homologous framed links in 3-manifolds
14:00 〜15:00
Takefumi Nosaka (Kyushu University)
Longitudes in SL2 -representation spaces of knot groups and Milnor-Witt K2 groups of ﬁelds
15:10 〜 16:10
Thang Le (Georgia Institute of Technology)
On the growth of torsions and regulators in ﬁnite abelian coverings
March 14
10:00 〜 11:00
Jun Murakami (Waseda University)
On the logarithmic knot invariants and the hyperbolic volume
11:10 〜 12:10
Rei Inoue (Chiba University)
Cluster Algebra and Complex Volume of Two-Bridge Knots1414:00 〜 14:40
Hirofumi Niibo (Kyushu University)
Idèlic class ﬁeld theory for 3-manifolds
14:50 〜 15:50
Gregor Masbaum (Institut de Mathématiques de Jussieu)
Integral TQFT and modular representations of mapping class groups

15

Low dimensional topology and number theory VI
March 18 - 21, 2014
Soft Research Park Center, Fukuoka
ProgramMarch 18
10:00 〜 11:00
Sakie Suzuki (Kyushu University)
The universal sl2 invariant and Milnor’s invariants
11:15 〜 12:15
Jochen Gärtner (University of Heidelberg)
Massey products and mild pro-p fundamental groups
14:00 〜 15:00
Naotake Takao (RIMS, Kyoto University)
On the monodromy representation associated to the universal family of algebraic curves
15:15 〜 16:15
Hiroyuki Fuji (Tsinghua University)
The colored HOMFLY homology and super-A-polynomial
March 19
10:00 〜 11:00
Kirsten Wickelgren (Harvard University)
Massey products in Galois cohomology via rational points
11:15 〜 12:15
Shinya Harada (Tokyo Institute of Technology)
Hasse-Weil zeta functions of A-polynomials of torus knots
14:00 〜 15:00
Ido Efrat (Ben Gurion University)
16

Filtrations of absolute Galois groups
15:15 〜 16:15
Megumi Hashizume (Nara Women’s University)
On the homomorphism induced by region crossing change
March 20
10:00 〜 11:00
Yuichiro Takeda (Kyushu University)
Higher arithmetic Chern character
11:15 〜 12:15
Thang Le (Georgia Institute of Technology)
On the skein modules of 3-manifolds at roots of unity
14:00 〜15:00
Anna Beliakova (Universität Zürich)
Trace as an alternative decategoriﬁcation functor
15:15 〜 16:15
Ken-ichi Sugiyama (Chiba University)
On a generalization of Deuring’s results
March 2110:00 〜 11:00
Akira Yasuhara (Tokyo Gakugei University)
Milnor Invariants and the HOMFLYPT Polynomial
11:15 〜 12:15
Yuriko Umemoto (Osaka City University)
The growth function of hyperbolic Coxeter dominoes and 2-Salem numbers
14:00 〜 15:00
Kazuo Habiro (RIMS, Kyoto University)
Kirby calculus for null-homologous framed links in 3-manifolds

17Abstract

Anna Beliakova, Trace as an alternative decategoriﬁcation functor
Categoriﬁcation is a lifting of a given mathematical structure to a higher categorical level. Decategoriﬁcation is the inverse process of simplifying higher
structure into the original one. Both procedures are not unique. Usually, theGrothendieck group K0 is used as a decategoriﬁcation functor. In this talk,
we illustrate on the example of categoriﬁed quantum groups that the trace or
0th Hochschild homology is an interesting alternative to K0 . We show that
duality between trace and center gives rise to an action of the current algebraUsl(n)[t] on the center of any 2-representation of the categoriﬁed quantum
sl(n). This was previously observed by Brundan for t = 1.
Hiroyuki Fuji, The colored HOMFLY homology and super-A-polynomial
I will discuss about the colored HOMFLY homology via string dualities. In
recent years, the string dualites are applied to the quantum knot invariants,
and some novel aspects of such invariants are studied explicitly in the physics
language. One of the main focus of my talk is the physical realization of thecategoriﬁcation of the colored HOMFLY polynomial that is named as thecolored HOMFLY homology. Utilizing string dualities, for some knots, wehave obtained the explicit formula for the Poincaré polynomial of this homology which is known as the colored superpolynomial. From such explicit
formulas, we further discovered the existence of the q-diﬀerence operator annihilating the colored superpolynomial for some knots, and it leads to a 2parameter extension of the A-polynomial in the semi-classical limit q → 1.
We named this extended A-polynomial as super-A-polynomial. This talk isbased on series of works in collaboration with S. Gukov, P. Sulkowski, and
M. Stošić.
Jochen Gärtner, Massey products and mild pro-p fundamental groups
In this talk we investigate the étale pro-p fundamental group π1ét (Spec Ok\S)(p), where S is a ﬁnite set of primes of the ring of integers Ok of a number
ﬁeld k. In the ﬁrst part we discuss arithmetic consequences of Spec Ok \ S
being of type K(π; 1) for p due to A. Schmidt and show how the ﬁrst explicit
18

examples in the diﬃcult tame case have been obtained by J. Labute using
the theory of mild pro-p groups. In the second part we recall the notion ofMassey products in the cohomology of pro-p groups. We give an application of Massey products in order to construct mild pro-p groups and discussarithmetic interpretations.
K. Wickelgren, Massey products in Galois cohomology via rational points
The Milnor conjecture identiﬁes the cohomology ring H∗ (Gal(k̄/k), Z/2Z)
with the tensor algebra of k ∗ mod the ideal generated by x ⊗ 1 − x forx ∈ k \ {0, 1} mod 2. In particular, x ∪ 1 − x vanishes, where x in k ∗ is
identiﬁed with an element of H1 . We show that order n Massey products ofn − 1 factors of x and one factor of 1 − x vanish by embedding P1 \ {0, 1, ∞}into its Picard variety and constructing Gal(k̄/k) equivariant maps from π1étapplied to this embedding to unipotent matrix groups. This also identiﬁes
Massey products of the form ⟨1 − x, x, ?, x, 1 − x⟩ with f ∪ 1 − x, where f
is a certain cohomology class which arises in the description of the action ofGal(k̄/k) on π1ét (P1 \ {0, 1, ∞}).

19

Low dimensional topology and number theory VII
March 25 - 28, 2015
IMAQ Seminar Room, Innovation Plaza, Momochihama, Fukuoka, JAPAN
ProgramMarch 25 (Wednesday)
11:00 〜 12:00
Thang Le (Georgia Institute of Technology)
On the quantum trace map in the theory of quantum Teichmuller spaces
14:00 〜 15:00
Takahiro Matsushita (The University of Tokyo)
Box complexes and Kronecker double coverings of graphs
15:15 〜 16:15
Yuri Berest (Cornell University)
Double aﬃne Hecke algebras and character varieties of knots
March 26 (Thursday)
10:00 〜 11:00
Akishi Kato (The University of Tokyo)
Quiver mutation loops and partition q-series
11:15 〜 12:15
Irina Davydenkova (Université de Genève)
Inequalities from Poisson brackets
14:00 〜 15:00
Tetsuya Ito (RIMS, Kyoto University)
A topological description of colored Alexander invariant
15:15 〜 16:15
Gwénaël Massyeau (Universitè de Strasbourg & CNRS)
A functorial extension of the Reidemeister torsion to the category of (2+1)dimensional cobordisms.

20

Banquet
March 27 (Friday)
10:00 〜 11:00
Kenji Sakugawa (Osaka Univeristy)
Polylogarithmic analogue of the Coleman-Ihara formula
11:15 〜 12:15
Anton Alekseev (Université de Genève)
The Grothendieck-Teichmueller group and the Kashiwara-Vergne problem
14:00 〜15:00
Kazuo Habiro (RIMS, Kyoto University)
Traces of categoriﬁed quantum groups
15:15 〜 16:15
Kentaro Okamoto (Kyushu University)
Properties of braid zeta function
March 28 (Saturday)
10:00 〜 11:00
Makoto Sakuma (Hiroshima University)
Mapping class group action on the space of geodesic rays
11:15 〜 12:15
Tsuyoshi Kobayashi (Nara Women’s University)
A construction of ﬂat-foldable origami via similarity structure21

Abstract

Anton Alekseev (Université de Genève)
The Grothendieck-Teichmueller group and the Kashiwara-Vergne problem
In this talk, we will explain the relation between the Kashiwara-Vergne problem in Lie theory and the Grothendieck-Teichmueller group. The KashiwaraVergne problem is stated in terms of the Campbell-Hausdorﬀ series and a certain equation for cyclic words in 2 variables. The Grothendieck-Teichmuellergroup gives rise to symmetries of the Kashiwara-Vergne problem. This observation leads to an interesting conjecture about the Grothendieck-Teichmueller
Lie algebra.
The talk is based on joint works with C. Torossian.
Yuri Berest (Cornell University)
Double aﬃne Hecke algebras and character varieties of knots
Let G be a complex reductive algebraic group. In this talk, we will discussa general conjecture that there is a canonical action of a double aﬃne Hecke
algebra of type G on a (quantized) character variety of a knot complementin S 3 . We will give some motivation and present an explicit constructionin the rank one case. We then discuss some examples and implications of
our conjecture. As a main application, we construct 3-variable polynomial
knot invariants that specialize to the classical colored Jones polynomials introduced by Witten, Reshetikhin and Turaev.
The talk is based on joint work with P. Samuelson.
Irina Davydenkova (Université de Genève)
Inequalities from Poisson brackets
We introduce the notion of tropicalization for Poisson structures on Rn with
coeﬃcients in Laurent polynomials. To such a Poisson structure we associatea polyhedral cone and a constant Poisson bracket on this cone. There is aversion of this formalism applicable to C n viewed as a real Poisson manifold.
In this case, the tropicalization gives rise to a completely integrable systemwith action variables taking values in a polyhedral cone and angle variables
spanning a torus.
As an example, we consider the canonical Poisson bracket on the dual PoissonLie group G∗ for G = U(n) in the cluster coordinates of Fomin-Zelevinsky
deﬁned by a certain choice of solid minors. We prove that the corresponding
22

integrable system is isomorphic to the Gelfand-Zeiltin completely integrable
system.
Kazuo Habiro (RIMS, Kyoto University)
Traces of categoriﬁed quantum groups
The trace (or cocenter of the 0th Hochschild-Mitchell homology) of a category is the space of endomorphisms modulo the relation f g = gf . The notionof trace can be naturally applied to categories with structures. For example,
the trace of a monoidal category is a monoid and the trace of a 2-category
is a (1-)category.
I plan to talk about generalities about trace and then the traces of categoriﬁed quantum groups.
Tetsuya Ito (RIMS, Kyoto University)
A topological description of colored Alexander invariant
The Alexander polynomial (invariant) is one of the most basic invariantof knots, and plays an important role in arithmetic topology. In this talkI present a homological description of the colored Alexander invariant, aquantum-invariant theoretical generalization of Alexander invariant. Our
description suggests a connection between an abelian covering of (the conﬁguration space of) the knot complement and partially explains that colored
Alexander invariant is indeed a generalization of the Alexander polynomial,in a topological point of view.
Akishi Kato (The University of Tokyo)
Quiver mutation loops and partition q-seriesThis is a joint work with Yuji Terashima (Tokyo Institute of Technology);
arXiv:1403.6569 & arXiv:1408.0444, Comm. Math. Phys. to appear.
Quivers (oriented graphs) and their mutations are ubiquitous in mathematics
and mathematical physics. They play a key role in wall-crossing phenomena,
gluing of ideal tetrahedra, etc.
Recently, we introduced a partition q-series Z(γ) for a quiver mutation loop γ(a loop in a quiver exchange graph in cluster algebra terminology). This enjoys following remarkable properties: (1) Z(γ) is invariant under “inversion”
and “cyclic shift” of γ; so it may be regarded as a monodromy invariant. (2)Z(γ) satisﬁes pentagon identities. (3) If the quivers are of Dynkin type orsquare products thereof, they reproduce so-called fermionic character formulas of certain modules associated with aﬃne Lie algebras; in particular they23

have nice modular properties. (4) If the mutation sequences γ is reddening,Z(γ) is expressed as a product of quantum dilogarithms and coincides withthe combinatorial Donaldson-Thomas invariant.
The deﬁnition of Z(γ) requires only combinatorial data of quivers and mutation loops, and completely independent of the details of the models behind.
It is hoped that a deeper understanding of the partition q-series shed newlights on dualities and quantization.
Tsuyoshi Kobayashi (Nara Women’s University)
A construction of ﬂat-foldable origami via similarity structure
Origami is a Japanese traditional artwork made of a square paper, and isstudied by many researchers, not only because it is theoretically interesting but also because it has lots of practical applications. In this talk, I
will describe a method for producing ﬂat-foldable origami by using similarity
structure on 2-dimensional torus, and Lorenz template. I will also mention
some topics related them.
Thang Le (Georgia Institute of Technology)
On the quantum trace map in the theory of quantum Teichmuller spaces
Takahiro Matsushita (The University of Tokyo)
Box complexes and Kronecker double coverings of graphs
Coloring of a graph is to assign a color to each vertex so that adjacent vertices
have diﬀerent colors. The chromatic number is the smallest number of colors
we need to color a given graph. The graph coloring problem is to determine
the chromatic number, and is one of the oldest problem in graph theory.
The box complex is a Z2 -space assigned to a graph, and its topology is known
to be closely related with the chromatic number. In this talk I related box
complexes with the canonical double coverings over graphs, called the Kronecker double coverings.
The main result states that if the Kronecker double coverings of two graphs
coincide then their box complexes are isomorphic. As an application, I constructed graphs G, H such that their box complexes are isomorphic but theirchromatic numbers are diﬀerent.
Gwénaël Massyeau (Universitè de Strasbourg CNRS)
A functorial extension of the Reidemeister torsion to the category of (2+1)dimensional cobordisms.
24

(Joint work with Vincent Florens.) In 3-dimensional topology, the Reidemeister torsion is a classical invariant which usually applies to manifolds
with empty or toroidal boundary. In this talk, we will consider Reidemeister
torsion for 3-manifolds with arbitrary boundary. Given a commutative ﬁeld
F and a multiplicative subgroup G of F , we will construct in this way afunctor from the category of (2+1)-dimensional cobordisms equipped with
G-representations to the category of graded F -vector spaces. We will show
a few properties of this functor and how it specializes to some well-known
Alexander-type invariants
Kentaro Okamoto (Kyushu University)
Properties of braid zeta function
There is a well-known zeta function of an element of the symmetric group asa toy model.
In this study, considering the extension from the symmetric group to thebraid group we introduce a new zeta function of braids by using the Burau
representation. We will show that this braid zeta function satisﬁes a functional equation and the analogue of Riemann Hypothesis under a certain
assumption, and that the Alexander polynomial is contained in the residueof this braid zeta function at s = 1. Furthermore, we deﬁne a special productas an operation of braids, and show a decomposition theorem of the zetafunction of a braid which is represented by special products. This theorem
leads to the decomposition formula of the Alexander polynomial of a special
knot as a corollary.
Kenji Sakugawa (Osaka Univeristy)
Polylogarithmic analogue of the Coleman-Ihara formula
Abstract: Let K be a number ﬁeld, z an element of K, p an odd prime andm a positive integer. The value ℓim (z) of the ℓ-adic polylogarithm of weightm at z is a continuous map from the absolute Galois group GK of K to the
GK -module
Qp (m). In this talk, we give a suﬃcient condition for the linear∑
sum i ai ℓim (zi ), ai ∈ Z, zi ∈ K being a 1-cocycle. Then, we give a formula
relating the restriction of that linear sum to a inertia subgroup of GK at aplace v over p and Coleman’s p-adic polylogarithms. This is joint work with
Hiroaki Nakamura and Zdzislaw Wojtkowiak.
Makoto Sakuma (Hiroshima University)
Mapping class group action on the space of geodesic rays of a punctured
25

hyperbolic surface
For a hyperbolic punctured surface S of ﬁnite area, consider the space, G, ofgeodesic rays emanating from punctures. Note that G is identiﬁed with the
disjoint union of the horo-cycles around the punctures and that the mappingclass group of S naturally acts on the space G.
In the ﬁrst half of my talk, I will explain the role of the action in the variations of McShane’s identity (i) for punctured surface bundles (Bowditch andAkiyoshi-Miyachi-Sakuma) and (ii) for 2-bridge links (Lee-Sakuma).
In the second half of my talk, I will explain the following theorem proved byBowditch, answering to a my question.
Theorem. The non-wandering set of the action of the mapping class groupon the space G has measure 0.
I would also like to discuss the mapping class group action on the space of”simple” geodesic rays.

26

Low dimensional topology and number theory VIII
March 22 - 25, 2016
IMAQ Seminar Room, Innovation Plaza, Momochihama, Fukuoka, JAPAN
ProgramMarch 22 (Tuesday)
9:20 〜 10:10
Kazuo Habiro (Kyoto University, RIMS)On the category of ﬁnitely generated free groups
10:25 〜 11:15
Thang Le (Georgia Institute of Technology)
On the q-holonomicity of the colored HOMFLYPT polynomial
11:30 〜 11:55
Kentaro Okamoto (Kyushu University)
The zeta function and Jones polynomial
14:30 〜 15:20
Ted Chinburg (University of Pennsylvania)
Negative curves on surfaces and hyperbolic codes
15:40 〜 16:30
Yasushi Mizusawa (Nagoya Institute of Technology)
Tame pro-p Galois groups over cyclotomic Zp -extensions
March 23 (Wenesday)
9:20 〜 10:10
Tomoki Mihara (University of Tokyo)
On M2 KR dictionary for metrised 3-dimensional manifolds
10:25 〜 11:15
Nigel Boston (University of Wisconsin-Madison)
Moments of distributions of algebraic fundamental groups
11:30 〜 11:55
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Hisatoshi Kodani (Kyushu University)
Group-like expansions, string links and some remarks on arithmetic topology
14:30 〜 15:20
Hiroshi Tsunogai (Sophia Univeristy)
Galois orbits of genus 1 dessins of degree up to 6
15:40 〜 16:30
Herbert Gangl (Durham University)
Zagier’s polylogarithm conjecture revisited
BanquetMarch 24 (Thursday)
9:20 〜 10:10
Shinya Harada (Tokyo Institute of Technology)
Hasse-Weil zeta functions of SL2 (PSL2 )-character varieties of closed orientable hyperbolic 3-manifolds
10:25 〜 11:15
Adam Sikora (State University of New York at Buﬀalo)
Algebraic Properties of Skein Algebras of Surfaces
11:30 〜11:55
Ryoto Tange (Kyushu University)
On certain L-functions for deformations of knot group representations
14:30 〜 15:20
Takao Satoh (Tokyo University of Science)
On the SL(m, C)-representation rings of free groups and the Johnson homomorphisms
15:40 〜 16:30
Anh Tran (The University of Texas at Dallas)
The topology of the Jones polynomial
March 25 (Friday)
9:20 〜 10:10
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Yu Iijima (Kyoto University, RIMS)
Galois action on the image of the pro-l universal monodromy representationof the moduli stack of hyperbolic curves
10:25 〜 11:15
Hidekazu Furusho (Nagoya University)
Complex and p-adic multiple zeta functions
14:00 〜 14:50
Nobuyoshi Takahashi (Hiroshima University)
Quandles associated to arithmetic schemes
15:10 〜 16:00
Alan Reid (The University of Texas at Austin)
Azumaya algebras and canonical components

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Abstract

Nigel Boston (University of Wisconsin-Madison)
Moments of distributions of algebraic fundamental groups
There are various important conjectures on distributions of Galois groupsof unramiﬁed extensions of imaginary quadratic ﬁelds. In this talk I shall
explain how these are more conveniently described in terms of moments andprove some function ﬁeld analogues. Joint work with Melanie Wood andothers.
Ted Chinburg (University of Pennsylvania)
Negative curves on surfaces and hyperbolic codes
This talk is about joint work with Matt Stover on curves of small genus andnegative self intersection on a smooth projective surface X. We show that
there are only ﬁnitely many such curves having arithmetic genus less than
one-fourth the ﬁrst Betti number of the surface; this result is sharp over ﬁnite ﬁelds. In the Neron Severi group of the surface, the classes of the above
curves form a counterpart in hyperbolic space of a spherical code. By bounding the number of points in such hyperbolic codes, we bound the number ofcurves of the above kind in terms of the rank of the Neron Severi group of
the surface.
Hidekazu Furusho (Nagoya University)
Complex and p-adic multiple zeta functions
This talk is based on my joint work with Y. Komori, K. Matsumoto and
H. Tsumura. In the ﬁrst half of my talk, I will introduce our method ofdesingularization of multiple zeta-functions. I will explain that multiple zetafunctions (which are known to be meromorphic with inﬁnitely many singularloci in the whole space) turn to be entire by our integral method. I will
introduce various properties of our methods: particularly I will reveal thatthe desingularized function is given by a suitable ﬁnite ’linear’ combinationof multiple zeta-functions with some arguments shifted. I will also explainthat speciﬁc combinations of Bernoulli numbers attain the special values attheir non-positive integers of the desingularized ones. Based on them, inthe second half, I will explain our construction of (several variable) p-adic
multiple L-functions which generalizes that of Kubota-Leopoldt’s p-adic Lfunctions. Then I will show their various fundamental properties including30

multiple Kummer congruences, functional relations as well as their evaluations at both positive and negative integer points.
Herbert Gangl (Durham University)
Zagier’s polylogarithm conjecture revisited
Abstract: In the early nineties, Goncharov proved the weight 3 case of Zagier’s Conjecture stating that the special value ζF (3) of a number ﬁeld F isessentially expressed as a determinant of trilogarithm values taken in thatﬁeld. He also envisioned a vast–partly conjectural–programme of how toapproach the conjecture for higher weight. We can remove one importantobstacle in weight 4 by solving one of Goncharov’s conjectures. It further
allows us to deduce a functional equation for Li4 in four variables as one
expects to enter in a more explicit deﬁnition of a certain algebraic K-group.
Kazuo Habiro (Kyoto University, RIMS)On the category of ﬁnitely generated free groups
I plan to give an elementary proof of the well-known fact that the category ofﬁnitely generated free groups is equivalent as a symmetric monoidal category
to the free symmetric monoidal category generated by a commutative Hopfmonoid. I plan to discuss also some related topics.
Shinya Harada (Tokyo Institute of Technology)
Hasse-Weil zeta functions of SL2 (PSL2 )-character varieties of closed orientable hyperbolic 3-manifolds
The SL2 (C)-character variety of a 3-manifold plays an important role in the
study of 3-dimensional topology, which is known to be an algebraic set over
the rational number ﬁeld. For an orientable hyperbolic 3-manifold of ﬁnite
volume we have an associated holonomy representation and there is an irreducible component of the SL2 (C)-character variety containing the characterof a lift of the holonomy representation (called a canonical component). In
this talk we will consider the closed hyperbolic 3-manifold case and explainthat the Hasse-Weil zeta function of the canonical component of the SL2 character variety (PSL2 -character variety in the sense of Heusener-Porti) of
a closed orientable hyperbolic 3-manifold of ﬁnite volume is equal to the
Dedekind zeta function of the trace ﬁeld (invariant trace ﬁeld). Here thetrace ﬁeld is a number ﬁeld generated by the trace of the image of the holonomy representation and the invariant trace ﬁeld is a subﬁeld of the trace ﬁeld
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generated by the trace of the square of the image. Especially we can see thatwhen the closed 3-manifold is an arithmetic 3-manifold, the special value ats = 2 of the zeta function of the PSL2 -character variety is expressed in termsof the hyperbolic volume of the manifold.
Yu Iijima (Kyoto University, RIMS)
Galois action on the image of the pro-l universal monodromy representationof the moduli stack of hyperbolic curves
By considering the outer action of the étale fundamental group of the moduli
stack of hyperbolic curves over Q on the pro-l fundamental group of a ﬁber ofthe universal curve over this moduli stack, we may obtain the pro-l universal
monodromy representation of the moduli stack of hyperbolic curves. This
representation may be regarded as a pro-l version of the natural outer actionof the mapping class group on the topological fundamental group of a connected orientable compact topological surface. The absolute Galois groupof Q acts naturally on the image of this representation up to inner automorphisms. In this talk, we study the structure of the image of the pro-l
universal monodromy representation by using the outer Galois action on thisimage.
Hisatoshi Kodani (Kyushu University)
Group-like expansions, string links and some remarks on arithmetic topology
In this talk, I introduce the notion of normalized expansions which satisﬁes
a boundary condition of a punctured disk and I give some applications ofthese expansions in the study of string links. When normalized expansions
are special expansions, we have the special Artin representation which gives
the special longitudes of a string link. I will also explain that the coeﬃcientsof a special longitude give a topological analogue of multiple zeta values insome sense.
Thang Le (Georgia Institute of Technology)
On the q-holonomicity of the colored HOMFLYPT polynomial
Tomoki Mihara (The University of Tokyo)
On M2 KR dictionary for metrised 3-dimensional manifolds
I develop M2 KR dictionary of arithmetic topology for branched Abelian coverings of metrised 3-dimensional manifolds such as 3-dimensional Riemannian
manifolds. I study analogy between rectiﬁable knots in metrised closed ori32

ented 3-dimensional rational homology spheres and prime ideals of number
ﬁelds. Since the class of rectiﬁable paths contains the class of geodesic paths,
the class of recitiﬁable knots generates the ﬁrst homology group, and hence
possesses suﬃcient data on Abelian coverings. The arc-length of rectiﬁable
knots corresponds to the cardinality of the residue ﬁelds, and the pairing
given by linking numbers yields an analogue of the notion of the reduction to
the residue ﬁeld. This correspondence is not just an extension of the analogy
between linking numbers modulo 2 and Artin symbols in class ﬁeld theory for
number ﬁelds, and plays an important role in the observation of local-global
compatibility of class ﬁeld theory for manifolds.
Yasushi Mizusawa (Nagoya Institute of Technology)
Tame pro-p Galois groups over cyclotomic Zp -extensions
For a prime number p and a number ﬁeld, we consider the maximal pro-pextension with restricted ramiﬁcation, which is tamely ramiﬁed over the cyclotomic Zp -extension. From a view of arithmetic topology, the Galois groupsof such extensions seem to be analogous to link groups, and Zp -extensions areanalogous to branched Z-covers of a link. An essential part of such a Galois
group is the Galois subgroup corresponding to the tamely ramiﬁed extension
over the Zp -extension. This tame pro-p Galois group has been studied in
Iwasawa theory with the Zp -action on it. In this talk, I will explain some
explicit examples of these Galois groups with some related topics.
Kentaro Okamoto (Kyushu University)
The zeta function and Jones polynomial
There is the well known zeta function of the dynamical system generated byan element of the symmetric group. By considering this zeta function as amodel, we construct the braid zeta function by using the Burau representation. Then the Alexander polynomial of the closure of a braid is interpreted
as the residue of this braid zeta function. Furthermore, we also construct
the zeta function by using the Jones representation of the braid group. In
this workshop, we talk about the relation between the zeta function and theJones polynomial.
Alan Reid (The University of Texas at Austin)
Azumaya algebras and canonical components
Abstract: Suppose that G is the fundamental group of a ﬁnite volume orientable 1-cusped hyperbolic 3-manifold M . Thurston identiﬁed a particular
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curve component C (the canonical component) of the character variety. This
component contains the faithful discrete character as well as those charactersarising from Dehn surgery. In this talk we will describe how one obtains a
so-called Azumaya algebra, A over a Zariski-open subset of C and deriveconditions on when A extends over all points of the smooth projective modelof C. As a consequence we will see how this puts signiﬁcant restrictions onarithmetic invariants of Dehn surgeries on M . In addition, when M is thecomplement of a hyperbolic knot K in S 3 , we can describe a suﬃcient condition for A to extend over C in terms of the Alexander polynomial of K.
Takao Satoh (Tokyo University of Science)
On the SL(m, C)-representation rings of free groups and the Johnson homomorphisms
First, we recall some facts about the Johnson homomorphisms and twistedﬁrst cohomology groups of the automorphism groups of free groups. Then
we introduce the rings of SL(2, C)-representations of free groups. By using
it, we can construct analogs of the Johnson homomorphisms. In this talk,we show that the ﬁrst homomorphism can be extended to the automorphismgroup of a free group as a crossed homomorphism, and show that it induces
Kawazumi’s 1-cocycle and Morita’s 1-cocycle of the automorphism group ofa free group.
Adam Sikora (State University of New York at Buﬀalo)
Algebraic Properties of Skein Algebras of Surfaces
For a surface F , the space of links in F × [0, 1] modulo Kauﬀman bracket
skein relations is called the skein algebra of F , denoted by S(F ). It is a
non-commutative deformation of the SL(2, C)-character variety of F , of signiﬁcant importance to quantum topology. In particular, for F with boundary,
it is (almost) the quantum Teichmuller space of F . Except for a few simplest
surfaces F , not much is known about the algebraic properties of S(F ) for
closed F . We are going to explore some of their two fundamental properties.
Nobuyoshi Takahashi (Hiroshima University)
Quandles associated to arithmetic schemes
A quandle is an algebraic system deﬁned by a binary operation. To a knot
K, its “knot quandle” Q(K) is associated, and Q(K) determines K up toequivalence. After a brief introduction to quandles, I will talk about a certain
analogue of the knot quandle in number theory: For a normal, separated and34integral scheme X of ﬁnite type over Z and a set M of closed points of X,
a topological quandle Q(X, M) is deﬁned. Then I will discuss the problem
of reconstruction. Let X = Spec OK \ {p} where K is Q or a quadraticﬁeld. Assume that π1 (X)ab is inﬁnite, and that M is of density 1. Then we
can recover pretty much information about X and M from Q(X, M) using
results from p-adic transcendental number theory.
Ryoto Tange (Kyushu University)
On certain L-functions for deformations of knot group representations
We study the twisted knot module for the universal deformation of an SL(2)representation of a knot group, and introduce an associated L-function, which
may be seen as an analogue of the algebraic p-adic L-function associated to
the Selmer module for the universal deformation of a Galois representation.
We then investigate two problems proposed by Mazur: Firstly we show thetorsion property of the twisted knot module over the universal deformation
ring under certain conditions. Secondly we verify the simplicity of the zeroesof the L-function by some concrete examples for 2-bridge knots. This is thejoint work with T. Kitayama, M. Morishita and Y. Terashima.
Anh Tran (The University of Texas at Dallas)
The topology of the Jones polynomial
I will discuss old and new conjectures about the topology of the Jones polynomial. These include the AJ conjecture, the slope conjecture, and the strong
slope conjecture. The AJ conjecture of Garoufalidis relates the A-polynomial
and the colored Jones polynomial of a knot. The A-polynomial was introduced by Cooper et al. in 1994 and has been fundamental in geometric
topology. A similar conjecture to the AJ conjecture was also proposed by
Gukov from the viewpoint of the Chern-Simons theory. The slope conjectureof Garoufalidis and two new conjectures of Kalfagianni and myself are about
the relationship between the degree of the colored Jones polynomial of a knot
and the topology of the knot. These conjectures assert that certain boundary
slopes and Euler characteristics of essential surfaces in a knot complement
can be read oﬀ from the degree of the colored Jones polynomial.
Hiroshi Tsunogai (Sophia University)
Galois orbits of genus 1 dessins of degree up to 6
We calculated the deﬁning equations of all Belyi pairs of genus 1 of degree
up to 6, and determined the Galois orbits in these cases. As a result, we35

show that these Galois orbits can be separated by known Galois invariantsof dessins: valency lists, monodromy groups, Nielsen classes, cartographic
groups and automorphism groups. We will give some examples of the computation of such Belyi pairs.

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