The 13th KOOK-TAPU Joint Seminar on Knots and Related Topics and
The 15th Graduate Student Workshop on Mathematics
Program of The 13th KOOK-TAPU Joint Seminar on Knots and Related Topics
July 26 (Tue)
9:20–9:30 Opening remarks(Chair: Wataru Yuasa)
9:30–10:00 Sam Nelson (Claremont McKenna College)
Biquandle Bracket Quivers
10:10–10:40 Jieon Kim (Pusan National University)
Symmetric biquandle cocycle invariants for unoriented links and surface-links(Chair: Sam Nelson)
10:50–11:20 Wataru Yuasa (OCAMI/RIMS)State-clasp correspondence for skein algebras
11:30–12:00 Seongjeong Kim (Jilin University)
On parity-like invariant for knots in Sg × S 1
July 27 (Wed)(Chair: Yeonhee Jang)
9:30–10:00 Seung Yeop Yang (Kyungpook National University)
Homology theories of distributive structures: computations
10:10–10:40 Erika Kuno (Osaka University)
Gromov hyperbolicity of nonseparating curve graphs for nonorientable surfaces(Chair: Seung Yeop Yang)
10:50–11:20 Byeorhi Kim (POSTECH)
On research related to 4-dimensional light bulb theorem
11:30–12:00 Yeonhee Jang (Nara Women’s University)
On keen bridge splittings
July 28 (Thu)(Chair: Sang Youl Lee)
9:30–10:00 Tetsuya Abe (Ritsumeikan University)
Ribbon concordance and the minimality of tight ﬁbered knots
10:10–10:40 Seonmi Choi (Kyungpook National University)
On marked graph mosaics(Chair: Tetsuya Abe)
10:50–11:20 Hirotaka Akiyoshi (Osaka Metropolitan University)
An extension of Ford domain
11:30–12:00 Sang Youl Lee (Pusan National University)
Dibiquandles and invariants for oriented dichromatic links
12:30–12:40 Closing remarks
1

Program of The 15th Graduate Student Workshop on Mathematics
July 26 (Tue)
13:20–13:30 Opening remarks
13:30–14:30 (Chair: Hirotaka Akiyoshi)
Ryo Takenaka (Osaka Metropolitan University)
Fermionic form of character formulas for aﬃne Lie algebras
Suhyeon Jeong (Pusan National University)
Invariants for pseudo-links derived from Goeritz matrices
Sadaf Habibi (Osaka City University)
Applications of p-harmonic transplantation for functional inequalities
Jinseok Oh (Kyungpook National University)
On set-theoretic Yang-Baxter cohomology of a certain family of Alexander biquandles
14:45–15:45 (Chair: Seongjeong Kim)
Minju Seo (Pusan National University)
A polynomial invariant for pseudo-links
Katsunori Arai (Osaka University)
Groupoid racks and spatial surfaces
Hongdae Yun (Kyungpook National University)
A geometric realization of the extreme Khovanov homology of a pretzel link
Hajime Kubota (Kyoto University)
The knot concordance invariant Υ using grid homology
16:00–17:00 (Chair: Byeorhi Kim)
Hyobeen Kim (Kyungpook National University)
The complexity dichotomy for matroid-homomorphism
Kanako Oie (Nara Women’s University)
Uniqueness of prime decomposition of 3-manifolds
Sanghoon Park (Pusan National University)
On the Gordon-Litherland pairing for links in thickened surfaces
Jumpei Yasuda (Osaka University)
A note on the plat index for surface-links
July 27 (Wed)
13:30–14:30 (Chair: Jieon Kim)
Jihyeon Lee (Pusan National University)
Rigidity theorems of λ-translating solitons
Unhou Kin (Nagoya City University)
pseudo Goeritz matrix of twisted knot
Hyungtae Baek (Kyungpook National University)
Some properties of S-strong Mori modules
Haruki Yoshimura (Osaka City University)
On a deformation of a cubical complex
2

14:45–15:45 (Chair: Erika Kuno)
Moemi Hiraki (Nagoya City University)
A virtualized skein relation for a multivariable polynomial under some conditions
Da Bin Park (Kyungpook National University)
Recovering of the initial values from light cone trace of solutions of the fractional wave equation
Fumika Mizoguchi (Osaka City University)
Nilpotent Lie algebras obtained by quivers and Ricci solitons
Youjin Lee (Pusan National University)
Data analysis and prediction of testing process in medical laboratory via machine learning
16:00–17:00 (Chair: Seonmi Choi)
Youn Jeong Oh (Kyungpook National University)
Graph Convolutional Neural Networks with modules that enable graph analysis more globally
Minami Matsumura (Nagoya City University)
An n-writhe for twisted knots
Jia Lijiao (Pusan National University)
Dynamic complexity of a three-dimensional host-parasitoids model with Logistic growth function
Yuta Taniguchi (Osaka University)
Surface knot invariants obtained from a quandle 2-cocycle

July 28 (Thu)
12:10–12:30 Awards Ceremony
12:30–12:40 Closing remarks

3Abstracts of The 13th KOOK-TAPU Joint Seminar on Knots and Related Topics
Tetsuya Abe (Ritsumeikan University)
Ribbon concordance and the minimality of tight ﬁbered knots
Agol proved that ribbon concordance forms a partial ordering on the set of knots in the 3-sphere.
In this talk, we prove that a tight ﬁbered knot, such as a torus knot, is minimal in the partially orderedset. This result gives a supporting evidence for a generalization of the slice-ribbon conjecture. This is
joint work with Keiji Tagami.
Hirotaka Akiyoshi (Osaka Metropolitan University)
An extension of Ford domain
The canonical decomposition of a noncompact hyperbolic manifold of ﬁnite volume with a single
cusp is deﬁned by Epstein and Penner. For 3-dimensional manifolds, it is used by the computer
software SnapPea due to Weeks, and known to be very much eﬀective to understand the properties ofmanifolds. The construction is applied to inﬁnite volume manifolds and the resulting decompositionis called the EPH-decomposition. It is known that the canonical decomposition (reps. a subcomplexof the EPH-decomposition) for a ﬁnite (resp. inﬁnite) volume manifold is dual to the Ford domain,which is another canonical fundamental domain. In this talk the deﬁnition of Ford domain is extended
to produce a convex set dual to the EPH-decomposition.
Seonmi Choi (Kyungpook National University)
On marked graph mosaics
In 2008, Lomonaco and Kauﬀman introduced a knot mosaic system to deﬁne a quantum knot
system. In 2014, Kuriya and Shehab proved Lomonaco-Kauﬀman conjecture which means that knot
mosaic type is a complete invariant of tame knots. The mosaic number of a knot K is the smallest
integer n for which K can be represented on an n × n mosaic board. In this talk, we deﬁne a mosaic
system for surface-links using marked graph diagrams and study the mosaic number of surface-links.
This talk is based on a joint work with Sam Nelson.
Yeonhee Jang (Nara Women’s University)
On keen bridge splittings
In this talk, we extend the concept of (strong) keenness for Heegaard splittings to bridge splittings,
and show that, for any integers n ≥ 1, g ≥ 0 and b ≥ 1 except for (g, b) = (0, 1), and (g, b, n) = (0, 3, 1),
there exists a strongly keen (g, b)-splitting of a link with distance n. We also show that any (0, 3)splitting of a link with distance 1 cannot be keen.
Byeorhi Kim (POSTECH)
On research related to 4-dimensional light bulb theorem
The classical light bulb trick states that a knot smoothly embedded in S 2 × S 1 that intersects a2
S × {y} transversally exactly one point is isotopic to the standard vertical curve. In 2020, David
Gabai generalized the classical light bulb trick to 4-dimensions. In this talk, we will give an overview
Gabai s 4-dimenstional light bulb theorem and other results introduced after Gabai. We also discuss
our result progress on isotopy of immersed 2-sphere.

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Jieon Kim (Pusan National University)
Symmetric biquandle cocycle invariants for unoriented links and surface-links
A quandle is a non-empty set with a binary operation with conditions derived from Reidemeistermoves. A quandle can be generalized to a biquandle, which is a non-empty set with two binary
operations. When we deﬁne invariants of (surface-)links by using quandles or biquandles, we use
orientations of (surface-)links. In this reason, it is diﬃcult to deﬁne invariants for non-orientablesurface-links by using (bi)quandles. In 2009, Kamada and Oshiro constructed symmetric quandles, aquandle with a map called a good involution, which give cocycle invariants for unoriented (surface)links.
In this talk, we deﬁne symmetric biquandles analogously to the symmetric quandles and introduce
symmetric biquandle (co)homology and symmetric biquandle cocycle invariants for unoriented links
and surface-links. This is a joint work with Sang Youl Lee.
Seongjeong Kim (Jilin University)
On parity-like invariant for knots in Sg × S 1
Because of the deﬁnition of ﬁbered links, one can obtain a knot K in ΣL × S 1 from a classical link
K ∪L, where L is a ﬁbered link and ΣL is a Seifert surface of L. K in ΣL ×S 1 has a geometric property
coming not only from the underlying surface ΣL but also S 1 . In this talk, we will discuss about knots
in Sg × S 1 where Sg is an oriented surface with genus g. We introduce basic notions for knots in
Sg × S 1 , for example, diagrams, moves for diagrams and so on. For knots in Sg × S 1 technically welose over/under information. But we will have information how many times the half of the knot at a
crossing rotates along S 1 . We introduce the parity-like invariant induced from the number of rotationof a half at a crossing and discuss about universal object for the parity-like invariant.
Erika Kuno (Osaka University)
Gromov hyperbolicity of nonseparating curve graphs for nonorientable surfaces
Curve graphs are often used to study mapping class groups of surfaces. A full subgraph of a curve
graph that consists of all vertices represented by the nonseparating curves is called the nonseparating
curve graph. After Masur–Minsky in 1999 proved that the curve graphs of orientable surfaces are
Gromov hyperbolic, various studies about the Gromov hyperbolicity for curve graphs have progressed.
In 2020, Rasmussen proved that the nonseparating curve graphs of orientable surfaces are Gromov
hyperbolic with a hyperbolicity constant which is independent of the topological types of orientablesurfaces. In this talk, we generalize the result by Rasmussen to nonorientable surfaces, and in particular
we would like to discuss about diﬀerences from the case of orientable surfaces.
Sang Youl Lee (Pusan National University)
Dibiquandles and invariants for oriented dichromatic links
A knot is a smooth simple closed curve in three-dimensional space and the disjoint union of n knots
is called an n-component link. A link with diﬀerent colors (labels) for each component is called adichromatic link. Usually, the two colors (labels) are represented by “1” and “2”. A special class ofdichromatic links is a class of links in the solid torus S 1 × D2 . So far, many invariants for classical links
have been generalized to those of dichromatic links. In this talk, we introduce an algebraic structure
called a dibiquandle which is a set equipped with four quandle operations satisfying certain conditions.
We discuss various examples and some properties of dibiquandles and also show that a dibiquandle
enables us to distinguish oriented dichromatic links by showing that the number of their all possible
colorings are diﬀerent when their arcs are colored by elements of the dibiquandle. This is a joint workof J. Kim and M. I. Sheikh.

5Sam Nelson (Claremont McKenna College)
Biquandle Bracket Quivers
Biquandle brackets are skein invariants for biquandle-colored oriented knots and links. Examples
include the classical skein invariants as well as the biuqandle and quandle 2-cocycle invariants. Biquandle coloring quivers are directed graph-valued invariants of oriented knots and links associated tosets of endomorphisms of a ﬁnite biquandle. Biquandle bracket quivers unite these two ideas to obtainan inﬁnite family of categoriﬁcations of biquandle bracket invariants, including new categoriﬁcationsof the Jones polynomial quite diﬀerent from Khovanov homology. This is joint work with Pia Cosma
Falkenburg.
Seung Yeop Yang (Kyungpook National University)
Homology theories of distributive structures: computations
Since C. S. Peirce emphasized the importance of self-distributivity in algebraic structures in 1880,
these structures have been actively studied by a number of scholars, including Mayer, Toyoda, Bruck,
Galkin, etc. While homology theories of associative structures, such as groups and rings, have been
extensively studied, starting with the work of Eileenberg and Hochschild, it has not been long since
the study of homology theories of non-associative distributed structures began to be active. In thistalk, we discuss what we are currently studying about homology theories of non-associative distributed
structures.
Wataru Yuasa (OCAMI/RIMS)State-clasp correspondence for skein algebras
For an oriented surface Σ with marked points M ⊂ ∂Σ and boundary intervals B, we introduce thestated g-skein algebra Sg (Σ, B) and the clasped g-skein algebra Sg (Σ, M) for g = sp4 . Moreover, weshow that the reduced version of the stated g-skein algebra Sgrd (Σ, B) is isomorphic to the boundarylocalization of the clasped g-skein algebra Sg (Σ, M)[∂ −1 ] for a Lie algebra g = sl2 , sl3 or sp4 . This
isomorphism is a quantum counterpart of the two descriptions of the cluster algebra associated withthe pair (g, Σ) in terms of the matrix coeﬃcients of Wilson lines and cluster variables, respectively.
This talk is based on a joint work with Tsukasa Ishibashi (Tohoku Univ.).

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Abstracts of The 15th Graduate Student Workshop on Mathematics
Katsunori Arai (Osaka University)
Groupoid racks and spatial surfaces
A rack is a set with a binary operation satisfying two conditions corresponding to two of the
Reidemeister moves in knot theory. We introduce the notion of a groupoid rack. It is used for coloringa spatial surface, which is a compact oriented surface with boundary embedded in S 3 . We show thata multiple group rack and a rack with a partially product deﬁned by Saito-Zappala are examples ofgroupoid racks.
Hyungtae Baek (Kyungpook National University)
Some properties of S-strong Mori modules
Let D be an integral domain, S a multiplicative subset of D and M a D-module. In 2014, Kim, Kim
and Lim deﬁned S-strong Mori modules; that is, every D-submodule of M is S-w-ﬁnite, equivalently,
if N is a D-submodule of M , then there exist s ∈ S and a ﬁnitely generated D-submodule L of N suchthat N s ⊆ Lw .
In this talk, we investigate some properties of S-strong Mori modules. To do this, we study thedeﬁnition of the star-operation; and w-operation which is one of examples of star-operations. The
main goal of this talk is Hilbert basis theorem for S-strong Mori modules. The detailed statement is
as follows.
Theorem (Hilbert basis theorem for S-strong Mori modules)
Let D be an integral domain, S an anti-Archimedean multiplicative subset of D and M a torsion-free
w-module over D. Then the following assertions are equivalent.
(1) M is an S-strong Mori D-module.
(2) M [X] is an S-strong Mori D[X]-module.
(3) M [X]Nv is an S-strong Mori D[X]Nv -module.
(4) M [X]Nv is an S-Noetherian D[X]Nv -module.

Sadaf Habibi (Osaka City University)
Applications of p-harmonic transplantation for functional inequalities involving a Finsler
norm
In this talk, we prove several inequalities such as Sobolev, Poincaré, logarithmic Sobolev, which
involve a general norm with accurate information of extremals, and are valid for some symmetricfunctions. We use Ioku’s transformation, which is a special case of p-harmonic transplantation, between
symmetric functions.
Moemi Hiraki (Nagoya City University)
A virtualized skein relation for a multivariable polynomial under some conditions
The virtualized skein relation for the Jones polynomial of virtual link diagrams was studied by
N. Kamada, S. Nakabo, and S. Satoh if one of diagrams in a virtual skein triple is checkerboard colorable. Also diﬀerent virtualized skein relations for the Jones polynomial were studied by N. Kamada.
On the other hand, H. A. Dye, L. H. Kauﬀman, and Y. Miyazawa intruoduced a multivariable polynomial invariant of virtual links, which is a reﬁnement of Jones polynomial. In this talk, we discussvirtual skein relations for the multivariable polynomial invariant.

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Suhyeon Jeong (Pusan National University)
Invariants for pseudo-links derived from Goeritz matrices
In 2010, Hanaki introduced the notion of a pseudo-diagram,which is a knot or link diagram where
we ignore over/under information at some crossings of the diagram. Henrich, Hoberg, Jablan, Johnson,
Minten, and Radvić extended this idea to a pseudo-knot (or pseudo-link ), which is an equivalenceclass of pseudo-diagrams modulo pseudo Reidemdister moves.
In this talk, we would like to introduce the Goeritz matrix for a pseudo-link, which is an extensionof the Goeritz matrix for a classical link. Using this, we show that the determinant of a modiﬁed
Goeritz matrix gives a Laurent polynomial invariant for pseudo-links in one variable u with integer
coeﬃcients. Also, we introduce the notions of the signature, determinant, and nullity of pseudo-links.
This is a joint work with Jieon Kim and Sang Youl Lee.
Hyobeen Kim (Kyungpook National University)
The complexity dichotomy for matroid-homomorphism
We introduce the diﬀerence between graph-homomorphisms and matroid-homomorphisms. With
this we prove a complexity dichotomy for the problem HomM (N) of deciding if there is a matroidhomomorphism to N . The problem is polynomial time solvable if N has a loop or has no circuitsof odd length, and is otherwise NP-complete. This is joint work with Mark Siggers at Kyungpook
National University and Cheolwon Heo at Sungkyunkwan University.
Unhou Kin (Nagoya City University)
pseudo Goeritz matrix of twisted knot
Twisted knot theory is an extension of virtual knot theory. Pseudo Goeritz matrices are deﬁned for
virtual knot diagrams. Torsion invariants of pseudo Goeritz matrices are invariants for virtual knots.
In this talk we discuss pseudo Goeritz matrices of twisted knots and their applications.
Hajime Kubota (Kyoto University)
The knot concordance invariant Υ using grid homology
Grid homology is an invariant for knots in S 3 . Grid homology eneble us to calculate some geometricinvariants for knots combinatorially. Two knots K1 , K2 ∈ S 3 are called concordant if there is a smooth
embedding of S 1 × [0, 1] into S 3 × [0, 1] having boundary the knots K1 and −K2 in S 3 × {0} andS 3 × {1}, respectively. The Υ invariant is a knot concordance invariant in knot Floer homology. In
this talk, we give a combinatorial restruction of the Υ invariant using grid homology and prove thatthe combinatorial Υ invariant is a concordance invariant.
Jihyeon Lee (Pusan National University)
Rigidity theorems of λ-translating solitons
In this talk, we are going to deal with λ-translating solitons in the Euclidean and Minkowski space.
A λ-translating soliton is a natural generalization of translating soliton, which is the equation of thelimit ﬂow by a proper blow-up procedure near type II singular points of the mean curvature ﬂow.
Thus, it is important to study (λ-)translating solitons for understanding the mean curvature ﬂow.
After some basic deﬁnitions and preliminaries of the λ-translating solitons in both spaces, in terms ofthe critical point of some area functional, we give the main results for classiﬁcations of λ-translating
solitons.

8

Youjin Lee (Pusan National University) Data analysis and prediction of testing process inmedical laboratory via machine learning
In the talk, I present our recent research on data analysis and prediction of testing process in medicallaboratory via machine learning to improve medical quality. Firstly, we addressed the importanceof laboratory results in medical ﬁelds and presented the current situation of work eﬃciency in thelaboratory. Next, we identiﬁed the causes that reduce the eﬃciency of test work. Finally, we proposed
an advanced integrated laboratory management system to solve the limitation of current system. It
supports machine learning-based test end time prediction, real-time TAT monitoring, and urgent test
request so that medical technologists can focus on their chemistry test working. Here, we employed
XGBoost, LightGBM, Random Forest, CatBoost regressions and their ensemble to predict the test
end time. As a result of the use in laboratories and outpatient clinics, the number of failures to comply
with the guided time and abnormal samples decreased.
Jia Lijiao (Pusan National University)
Dynamic complexity of a three-dimensional host-parasitoids model with Logistic growth
function
Host-parasitoid interactions have been studied for a long time, constructing host-parasitoid modeland exploring its stability have signiﬁcant meaning for biological control, which prefer the interaction
between pest and its parasitoid remain suﬃciently stable in order to prevent the host from re-emerging
as a pest. Under the natural environment, it is very common to ﬁnd an insect species being attacked by
two or more speciﬁc insect parasites. The problem we investigate in this study is mainly about a threedimensional host-parasitoid-parasitoid model with Logistic growth function on the host population. We
demonstrate the existence of boundary and positive equilibrium points of the three-dimensional system.
Under certain conditions, the positive equilibrium remains locally asymptotically stable. Moreover, it
is demonstrated that the system undergoes Neimark-Sacker bifurcation by using bifurcation theory. In
order to stabilize the unstable steady state, the pole-placement feedback control strategy is introduced.
Complexity and chaotic behaviour are conﬁrmed through the plots of bifurcation diagrams, maximum
Lyapunov exponents and phase portraits in numerical simulations.
Minami Matsumura (Nagoya City University)
An n-writhe for twisted knots
A double covering diagram from a twisted link diagram was introduced N. Kamada and S. Kamada.
S. Satoh and K. Taniguchi introduced the n-writhe, a virtual knot invariant for each non-zero integer
n. In this talk, We discuss n-writhes for double covering diagram from a twisted link diagram.
Fumika Mizoguchi (Osaka City University)
Nilpotent Lie algebras obtained by quivers and Ricci solitons
A quiver is a directed graph where loops and multiple arrows between two vertices are allowed. In
this talk, we introduce a new method for obtaining nilpotent Lie algebras from quivers. We consider
Lie algebras obtained by quivers without cycles, and prove that the corresponding simply connected
Lie groups always admit left-invariant Ricci solitons.

9Jinseok Oh (Kyungpook National University)
On set-theoretic Yang-Baxter cohomology of a certain family of Alexander biquandles
A homology theory of set-theoretic Yang-Baxter operators was established by Carter, Elhamdadi,
and Saito. Biquandles, a generalization of quandles, are special solutions of the set-theoretic YangBaxter equation. The free parts of quandle homology groups were completely determined, but only
little is known about biquandles. In this talk, we ﬁrst review the deﬁnition of set-theoretic Yang-Baxter(co)homology and determine the Betti numbers of some ﬁnite Alexander biquandles. This is joint workwith Seung Yeop Yang and Hongdae Yun.
Youn Jeong Oh (Kyungpook National University)
Graph Convolutional Neural Networks with modules that enable graph analysis moreglobally
Graphs have ﬂexible data structures that can be generalized to a variety of data structures. Accordingly, there is an advantage in that the problem can be viewed from a new perspective when
expressing data in a graph structure. We introduce Graph Convolutional Neural Networks, whichapplies the convolution neural networks model to graph data, one of the methods for analyzing graphs.
In particular, in order to create a GCN model that can analyze the graph structure globally as well
as locally, I studied with interest how to add a module that can explicitly know graph information sothat it can be analyzed in much more detail.
Kanako Oie (Nara Women’s University)
Uniqueness of prime decomposition of 3-manifolds
It is known that every orientable closed 3-manifold can be decomposed into a ﬁnite number ofprime orientable closed 3-manifolds by connected sum, and such decomposition is unique. In fact, theexistence of such decomposition is proved by H. Kneser, and the uniqueness is proved by J. Milnor. In
this talk, I introduce an outline of the proof of the uniqueness presented in the textbook of K. Morimoto.
Da Bin Park (Kyungpook National University)
Recovering of the initial values from light cone trace of solutions of the fractional wave
equation
We reconstruct the initial functions from the trace of solution of an initial value problem for thewave equation on the light cone. When the spatial dimension is odd, Rakesh and Yuan have alreadyprovide the inverting the operator trace the solution on the light cone from the initial function [1]. We
generalize their work to the fractional wave equation and all dimensions. That is, we present how toreconstruct the initial functions from the solution of the fractional wave equation on the light cone.
Sanghoon Park (Pusan National University) On the Gordon-Litherland pairing for links inthickened surfaces
In 1978, Gordon and Litherland deﬁned a symmetric, bilinear form which is called GordonLitherland pairing for links in S 3 together with a choice of unoriented spanning surface. Boden and
Karimi extended the pairing to links in thickened surfaces. Using it, they deﬁned some invariants signature, determinant, and nullity invariants for links in thichened surfaces. In this talk, based on
their paper, I review the Gordon-Litherland pairing for links in thickened surfaces.

10Minju Seo (Pusan National University)
A polynomial invariant for pseudo-links
A pseudo-diagram is a knot diagram that may be missing some crossing information. At thesecrossings, it is undetermined which strand passes over. Pseudo-link is an equivalence class of pseudodiagrams upto local moves of pseudo-diagrams, called pseudo-Reidemeister moves. Pseudo-diagrams
were ﬁrst introduced by Ryo Hanaki in 2010. In this talk, we introduce a modiﬁcation of the Jones
polynomial for pseudo-links. By using this modiﬁcation, we deﬁne a polynomial for pseudo-diagramsand discuss that the polynomial is invariant under pseudo-Reidemeister moves.
Ryo Takenaka (Osaka Metropolitan University)
Fermionic form of character formulas for aﬃne Lie algebras
Let g be a simple Lie algebra with an automorphism ν. The corresponding aﬃne Lie algebra isdenoted by g̃. In this talk, we write character fomulas for an aﬃne Lie algebra in terms of q series.
For some character formula, these fermionic character formulas are rewriting of Weyl-Kac character
formula written by formal exponentials. As a consequence, we are able to ﬁnd some connection to
Rogers-Ramanujan identities.
Yuta Taniguchi (Osaka University)
Surface knot invariants obtained from a quandle 2-cocycle
J. S. Carter, M. Saito and S. Satoh deﬁned an invariant of oriented surface knots using a quandle 2cocycle θ. In this talk, we relate the surface knot invariant using a quandle 2-cocycle θ to an f -twisted
Alexander matrix, which was introduced by A. Ishii and K. Oshiro, using the Alexander pair associatedwith the quandle 2-cocycle θ. As an application, we determined the second quandle homology groupof the knot quandle of 2-knots.
Jumpei Yasuda (Osaka University)
A note on the plat index for surface-links
A plat form for links is a presentation of a classical link using a braid. We can apply this presentation
to surface-links, using a braided surface instead of a braid, and prove that every surface-link has a platform presentation. The plat index is a surface-link invariant deﬁned by using a plat form and is ananalogy of the bridge index for classical links. In this talk, we compute the plat index of a 2-twist-spun
trefoil. Also, for any positive integer m, we give inﬁnitely many surface-knots with the plat index m.
Haruki Yoshimura (Osaka City University)
On a deformation of a cubical complex
It is known that the complement of a reduced alternating link admits a non-positively curved
cubing. I will talk about an attempt to obtain the hyperbolic structure as a deformation of the nonpositively curved structure. In particular, we focus on the two examples, the Borromean rings and theﬁgure-eight knot.
Hongdae Yun (Kyungpook National University)
A geometric realization of the extreme Khovanov homology of a pretzel link
The Khovanov (co)homology was introduced by Mikhail Khovanov in 2000, and Viro explained it interms of enhanced states of a link diagram. J. González-Meneses, P.M.G. Manchón, M. Silvero showedthe (potential) extreme Khovanov homology of a link is isomorphic to the independence simplicial
complex of a Lando graph from the link. In this talk, we investigate the extreme part of Khovanov
homology of certain pretzel links and their geometric realizations. This is joint work with Jinseok Oh
and Seung Yeop Yang.

11