Workshop
Knot theory, LMO invariants and related topics
at Okinawa Institute of Science and Technology (OIST)
1919-1 Tancha, Onna-son, Kunigami-gun, Okinawa, Japan 904-0495
March 9 - 11, 2024

1Schedule

10:00-10:30
Tea time
10:45-11:15
Break
11:30-12:00
Lunch
13:30-14:15
Tea time
14:30-15:00
Break
15:15-15:45
Tea time
16:00-17:00

2

March 9March 10

March 11Suzuki

10:00 - 10:45 Nozaki

Kitayama

Bao

11:00 - 11:45 Cha

ShengIshikawa

PetrouNozaki

Excursion

ChaKitano

SakasaiOzawa

Morifuji

discussion

discussion

Title and Abstrct1. Yuanyuan Bao (Tohoku University)
Title: gl(1|1) Alexander polynomial, Reidemeister torsion and lens space
Abstract: Based on nilpotent representations of quantum sl(2) at a root of unityof order 2r (r ≥ 2, not divisible by 4), Blanchet, Costantino, Geer and PatureauMirand constructed a series of TQFTs. They showed that the 3-manifold invariantof the r = 2 case is a canonical normalization of Reidemeister torsion. In this talk,we show that the 3-manifold invariant constructed from quantum gl(1|1) is also
equivalent to Reidemeister torsion, and we discuss its value for lens spaces.
2. Jae Choon Cha (POSTECH)
Minicourse: “Topological = smooth” in dimension 41

Abstract: In dimension 4, the works of Freedman and Donaldson led us to thestriking discovery that the smooth (or piecewise linear) category behaves very differently from the topological category, compared to other dimensions. Since then,it has been a very successful direction to investigate the diﬀerence between the twocategories in various contexts in dimension 4, using modern smooth invariants. On
the other hand, little was understood, if any, about when smooth and topological are similar in dimension 4. In this talk, we will discuss some recent progresson new topological = smooth results in dimension 4. In particular, we showthat topological and smooth isotopy classes of properly embedded disks admitting
algebraic duals in the boundary are bijective.
3. Katsumi Ishikawa (Kyoto Univesity)
Title: The trapezoidal conjecture for knots and links of braid index 3
Abstract: The trapezoidal conjecture is a classical famous conjecture posed byFox, which states that the coeﬃcient sequence of the Alexander polynomial of
any alternating knot is trapezoidal. In this talk, we show this conjecture for any
alternating links obtained by closing 3-braids.
4. Teruaki Kitano (Soka University)
Title: Some applications of a twisted Alexander polynomial to a symmetric union
presentation of a knot
Abstract: This is based on joint works with Michel Boileau (Marseille) and Yuta
Nozaki (Yokohama). A symmetric union presentation of a knot was originallystudied by Kinoshita-Terasaka as a union of knots in 1958. In 2000 Lamm deﬁneda symmetric union of even type or a skew-symmetric union. It is well known that
the Alexander polynomial of a knot K has a special form if K has a symmetricunion of even type. In this talk, we discuss that it can be generalized to a twisted
Alexander polynomial. We show some applications to the knot genus, the existenceof an epimorphism between knot groups, or relation with a ribbon knot as long as
one has time.
5. Takahiro Kitayama (University of Tokyo)
Title: Blanchﬁeld pairings and Gordian distance
Abstract: We present a lower bound of the Gordian distance of knots in terms of
their Blanchﬁeld pairings, extending a theorem of Borodzik and Friedl on the unknotting number of a knot. We also describe computational results, in particular,
giving examples of pairs of knots with the Gordian distance equal to 3 which are
diﬃcult to treat otherwise. This talk is based on joint work with Stefan Friedl andMasaaki Suzuki.
6. Takayuki Morifuji (Keio University)
Title: A volume presentation of a hyperbolic ﬁbered knot
Abstract: In this talk, we present a volume presentation of a hyperbolic ﬁberedknot using the Bell polynomials. More precisely, we show that the hyperbolic2

volume of a ﬁbered knot can be expressed in terms of the traces of powers of amonodromy matrix.
7. Yuta Nozaki (Yokohama National University)
Mini course: On the LMO invariant of 3-manifolds
Abstract: In the realm of quantum topology, various invariants are studied inthe framework of ﬁnite-type invariants. Le, Murakami and Ohtsuki introduced
the LMO invariant of closed 3-manifolds, which is universal among certain ﬁnitetype invariants. Subsequently, this invariant has been extended to 3-manifolds
with boundary and applied to the study of mapping class groups of surfaces. In
these talks, I will provide background on the LMO invariant and discuss recent
developments in related ﬁelds. These talks are partially based on joint work with
Masatoshi Sato and Masaaki Suzuki.
8. Yuko Ozawa (Meiji University)
Title: Epimorphisms between genus two handlebody-knot groups
Abstract: For two prime knots, a relation that there exists an epimorphism between
these knot groups is deﬁned. This relation has been determined up to elevencrossings. In this talk, we deﬁne a relation for handlebody-knots in the same way
and consider the relation for genus two handlebody-knots up to six crossings.
9. Andreani Petrou (OIST)
Title: Factorisation of the Harer-Zagier transform of the HOMFLY-PT polynomial
Abstract: I will talk about the Laplace transform of the HOMFLY-PT polynomial for knots, called the Harer-Zagier (HZ) transform, which is a function of twovariables λ and q. For some special families of knots it admits a fully factorised
form. This is not true, however, for the majority of knots, for which it can only
be decomposed as a sum of factorised terms. An interesting relation between thisdecomposition and Khovanov homology will be discussed. Notwithstanding, wesuggest that by ﬁxing the variable λ = q n , for some ”magical” exponent n, theHZ transform of any knot can obtain a factorised form in terms of cyclotomic
polynomials. Moreover, I will talk about the zeros of HZ transform which show aninteresting behavior.
10. Takuya Sakasai (University of Tokyo)
Title: Invariants of homologically ﬁbered knots
Abstract: A homologically ﬁbered knot is an enlargement of a ﬁbered knot. We
discuss invariants of homologically ﬁbered knots using Johnson-Morita theory. In
particular, we see some distinction between the genus 1 case and higher.
11. Xiaobing Sheng (OIST)
Title: On sequences of alternating links constructed from elements of Thompson’sgroup F
Abstract: Thompson’s groups F, T and V were ﬁrst constructed by Richard Thompson from a logical context where these groups were later found to have connections
3with many other branches of not only mathematics but also theoretical physics.
Jones, in his late years, motivated by an attempt to develop new conformal ﬁeld
theory, obtained unitary representations of Thompson’s groups F and T . as oneof the results he discovered a rather concrete way of constructing knots and links
from Thompson’s group F as an analogue of results on braid groups.
In this talk, we focus on a particular sequence of elements of the group F that
produces a sequence of alternating knots and we provide several diﬀerent ways
to look at the sequence through a selective of computational results. This is anongoing joint work with Bao Yuanyuan.
12. Massaki Suzuki (Meiji University)
Title: Twisted Alexander polynomials of knots associated to the regular representations of ﬁnite groups
Abstract: The twisted Alexander polynomial of a knot is deﬁned associated to alinear representation of the knot group. If there exists a surjective homomorphismof a knot group onto a ﬁnite group, then we obtain a representation of the knot
group by the composition the surjective homomorphism and the regular representation of the ﬁnite group. In this talk, we provide several formulas of the twisted
Alexander polynomial of a knot associated to such representations.

3Organizers

Shinobu Hikami (OIST), Masaaki Suzuki (Meiji University), Xiaobing Sheng (OIST)

4