The 16th TAPU-KOOK Joint Seminar
on Knots and Related Topics and
The 18th Graduate Student Workshop on Mathematics

Title : Finding Dirichlet domains, II

Speaker : Hirotaka Akiyoshi (Osaka Metropolitan University)

Abstract : In my talk at The 14th TAPU-KOOK Joint Seminar on Knots and Related Topics, I introduced a project involving numerical experiments whose aim is to determine the Dirichlet domains for hyperbolic manifolds. This talk is a continuation of that work, focusing on new developments and applications. We study Dirichlet domains for certain hyperbolic structures on the closed orientable surface of genus 2, and for every point on the surface we numerically determine the Dirichlet domain with respect to the point. Based on the result, we also discuss the diameters of the surfaces.

Title : Invariants for knots in $\Sigma_g\times I$ and knots in $\Sigma_g\times S^1$

Speaker : Zhiyun Cheng (Beijing Normal University)

Abstract : In this talk, I will discuss how to use index and quandle to define knot invariants for knots in 3-manifolds mentioned in the title. This talk is based on joint work with Hongzhu Gao.

Title : Cellular Transformer: A Topological Deep Learning Framework for Molecular Understanding

Speaker : Mustafa Hajij (University of San Francisco)

Abstract : We introduce the Cellular Transformer, a novel topological deep learning (TDL) framework that generalizes graph transformers to operate on regular cell complexes enabling a deeper and more expressive modeling of higher-order molecular structures. Accurately representing complex biomolecules remains a central challenge in computational chemistry, largely due to the intricate interplay between geometry and topology. While traditional graph neural networks (GNNs) and graph transformers have made strides, they often fall short in handling higher-order structures such as fused rings, aromatic systems, and nested cycles. These limitations arise from their reliance on node-edge representations and often require heuristics like virtual nodes, graph rewiring, or domain-specific encodings. Our approach is principled and native: CT directly leverages the structure of cell complexes to encode topological features—capturing loops, cliques, and ring-level motifs in a mathematically grounded manner. To support this, we introduce the augmented molecular cell complex, a richer molecular representation that retains essential ring- and bond-level context, critical for tasks in drug discovery and chemical property prediction. Extensive evaluations on MoleculeNet and newly constructed cell-complex datasets demonstrate that CT consistently outperforms standard GNNs and graph transformers, while avoiding the need for domain-specific augmentation. Remarkably, our model achieves this without relying on virtual nodes, structural encodings, or rewiring techniques, illustrating the power of topologically informed attention mechanisms in capturing subtle and global molecular interactions.

Title : Sah-Arnoux-Fathi invariants, algebraic K-theory and foam cobordisms

Speaker : Mee Seong Im (Johns Hopkins University)

Abstract : I will introduce scissor congruence and algebraic K-theory and how they are related to cobordism groups of foams in various dimensions. The former is joint with Mikhail Khovanov while the latter is joint with David Gepner, Mikhail Khovanov, and Nitu Kitchloo.

Title : Keenness for bridge splitting of links

Speaker : Yeonhee Jang (Osaka Metropolitan University)

Abstract : Hempel distance for a bridge splitting of a link is defined to be the distance in the curve complex of the bridge surface between the two subcomplexes corresponding to the two parts of the bridge splitting. We say that a bridge splitting is keen if the pair of vertices in the curve complex realizing the Hempel distance is unique, and is weakly keen if the set of the pairs of vertices realizing the Hempel distance is finite. In this talk, we review our result on the existence of keen bridge splttings and give some observations on weakly keen bridge splittings.

Title : On commutator identities obtained from doodles on a surface

Speaker : Seiichi Kamada (The University of Osaka)

Abstract : R. Fenn and P. Taylor introduced the notion of a doodle and showed that a doodle diagram, which is a collection of curves on the 2-sphere, induces an identity among commutators in a free group. This idea was observed more closely in a paper by A. Bartholomew, R. Fenn, N. Kamada and S. Kamada [JKTR 31 (2022)]. On the other hand, the notion of a doodle on the sphere can be extended to any surface with positive genus (cf.[JKTR 27 (2018)]). We discuss commutator identities obtained from doodles on a surface. It is a part of a project by A. Bartholomew, R. Fenn, N. Kamada and S. Kamada.

Title : Foams and their applications in link homology and categorification

Speaker : Mikhail Khovanov (Johns Hopkins University)

Abstract : In this series of talks we’ll introduce foams and their evaluation due to Robert and Wagner given by a version of state sum model in three dimensions that counts embedded surfaces. In the special 3-color case this model relates to the Four-Color Theorem and it also gives rise to a categorification of the quantum SL(3) invariant. We also explain how it works for N colors and discuss its applications, including to a 3-dimensional interpretation of the category of Soergel bimodules.

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Speaker : Byeorhi Kim (POSTECH)

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Title : Immersed surface-links presented by a triangle chs-graph

Speaker : Jieon Kim (Pusan National University)

An immersed surface-link is a surface smoothly immersed in $\Bbb R^4$ such that the multiple points are transverse double points. Such a surface can be represented by a planar 4-valent graph embedding with the vertices of three types. By considering the dual graph of this diagram, one can construct another presentation, called a chs-graph. In this talk, we focus especially on the case where the chs-graph takes the form of a triangular-shaped graph.

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Speaker : Yoosik Kim (Pusan National University)

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Title : Quiver Categorification of Homset Invariants

Speaker : Sam Nelson (Claremont McKenna College)

Abstract : The quandle coloring quiver and similar constructions provide a way of turning homset- based invariants into categories. In this talk we will examine this phenomenon with several examples.

Title : The intersection polynomials of a long virtual knot

Speaker : Kodai Wada (Kobe University)

Abstract : In this talk, we introduce twelve kinds of polynomial invariants of a long virtual knot called the intersection polynomials, and establish some fundamental properties of them such as their behavior under orientation-reversal/mirror-reflection, and additivity under concatenation product. Moreover, we provide a characterization of each of the intersection polynomials. We also introduce two kinds of supporting genera of a long virtual knot as a new topological property, and give a relationship between these supporting genera and the intersection polynomials. This is joint work with Takuji Nakamura, Yasutaka Nakanishi and Shin Satoh.

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Speaker : Xiao Wang (Jilin University)

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Speaker : Seung Yeop Yang (Kyungpook National University, Republic of Korea)

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Title : 2-plat 2-knots and their double branched covers

Speaker : Jumpei Yasuda (Osaka Metropolitan University)

Abstract : A 2-knot is a 2-sphere smoothly embedded in the 4-sphere. In this talk, we introduce a new family of 2-knots, called 2-plat 2-knots, which are higher-dimensional analogues of 2-bridge knots. These 2-knots admit normal forms parametrized by rational numbers. We provide a formula for their Alexander polynomials and determine their double branched covers.

Title : An infinite family of pairs of spatial surfaces

Speaker : Katsunori Arai (The University of Osaka)

Abstract : A spatial surface is a compact surface embedded in the 3-sphere. In this talk, we assume that spatial surfaces are oriented and that each connected component is neither a 2-disk nor a closed surface. Such surfaces can be presented by diagrams of spatial trivalent graphs and Reidemeister type theorem holds. A rack is a set equipped with a binary operation satisfying axioms motivated by two of the three Reidemeister moves in knot theory. A multiple group rack (MGR) is a rack defined on a disjoint union of groups, satisfying conditions corresponding to Reidemeister moves for spatial surfaces. Using MGRs, we can construct invariants of spatial surfaces up to ambient isotopy. In this talk, we provide an infinite family of pairs of spatial surfaces which have the same boundary, isomorphic fundamental groups of their complements, and Seifert matrices that are unimodular congruent. We show that the spatial surfaces in each pair of the family are not ambiently isotopic by distinguishing them using invariants derived from MGRs.

Title : Fractional-Order Modeling of HIV-Tuberculosis (TB) Co-infection with TB Prophylaxis in South Korea

Speaker : Amare Tezera Dessie (Pusan National University)

Abstract : Tuberculosis (TB) remains a significant public health challenge in South Korea, while HIV prevalence is relatively low. This ongoing study develops a fractional-order differential equation model tailored to the South Korean epidemiological context to investigate the dynamics of HIV-TB co-infection. The model incorporates TB treatment strategies, including Isoniazid Preventive Therapy (IPT) administered to individuals at risk of TB before infection to prevent progression from latent to active disease, and standard TB treatment regimens for active cases. Additionally, it quantifies the impact of Antiretroviral Therapy (ART) for HIV management in co-infected individuals. Fractional-order modeling is employed to capture memory effects and non-local interactions, providing a more accurate representation of complex biological and epidemiological processes compared to traditional integer-order models. Through numerical simulations, I assess the combined effectiveness of pre-infection IPT, postinfection TB treatment, and ART in reducing TB transmission, reactivation, and HIV progression. The study evaluates the stability and sensitivity of co-infection dynamics under various treatment scenarios. The fractional-order approach enhances predictive accuracy for treatment outcomes, offering critical insights for optimizing integrated TBHIV intervention strategies in South Korea, with a focus on HIV control.

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Speaker : Jinyoung Heo (Kyungpook National University)

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Title : Investigation on inflammation and remission times in atopic dermatitis system

Speaker : Jaewoo Hwang (Kyungpook National University)

Abstract : Atopic dermatitis (AD) is a chronic skin disease characterized by recurrent cycles of inflammatory response and remission, depending on the patient’s skin permeability ($\kappa_P$) and immune response ($\alpha_I$). Although it is not life-threatening, AD significantly impairs patients’ quality of life due to persistent or frequently recurring symptoms. In this talk, we discuss mathematical analysis of inflammation time ($\tau_I$) and remission time ($\tau_R$) in the AD system. We investigate how $\kappa_P$ and $\alpha_I$ affect $\tau_R$ and $\tau_I$ in cases of mild and serious oscillation, respectively. Especially, we highlight two interesting dynamical behaviors for these measurements: the logarithmic scaling law in $\tau_R$ and the phase transition of $\tau_R$ and $\tau_I$.

Title : An example of a family of quandle extensions

Speaker : Ryoya Kai (Osaka Metropolitan University)

Abstract : A quandle extension is a surjective quandle homomorphism whose fibers have the same cardinality. This notion is a generalization of a quandle 2-cocycle, and plays an important role in knot theory. However, it is difficult to construct quandle extensions in general. One of the simplest non-trivial examples is the extension from the octahedral quandle with type 4 to the dihedral quandle with degree 3. This example can be understood using natural projection from the 2-sphere to the projective plane. In this talk, we generalize this extension, and give a new infinite family of quandle extensions by using the geometry of oriented 2-plane Grassmannians, which is a generalization of the 2-sphere. In addition, we provide some properties of these extensions. This talk is based on a joint work with Hiroshi Tamaru (Osaka Metropolitan University).

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Speaker : Miji Kim (Pusan National University)

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Title : Nilpotent Lie algebras obtained by quivers with cycles and Algebraic Ricci solitons

Speaker : Hina Kitayama (Osaka Metropolitan University)

Abstract : A quiver is a directed graph represented by edges and vertices that allows multiple edges and loops. Recently, Mizoguchi-Tamaru developed a method for constructing nilpotent Lie algebra from quivers, and proved the nilpotent Lie algebras obtained from a quiver without cycles always admits an algebraic Ricci soliton. In this talk, I will first introduce a method for constructing nilpotent Lie algebras from quivers with cycles, which is an extension of the method from previous research. Then, I will present the results we have obtained regarding the conditions for the simply connected Lie groups corresponding to the obtained nilpotent Lie algebras to admit left-invariant Ricci soliton Riemannian metrics.

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Speaker : Hyungu Lee (Kyungpook National University)

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Speaker : Sangsoo Lee (Kyungpook National University)

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Speaker : Meiling Li (Pusan National University)

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Title : Asymptotic stability for a linear difference equation with off-diagonal delays

Speaker : Ryohei Mashino (Osaka Metropolitan University)

Abstract : The zero solution of a difference equation is called asymptotically stable if any solution converges to 0. The effect of delays on the asymptotic stability of the zero solution is fully understood only in some linear equations and is generally unsolved. In this study, we consider a linear difference equation with delays in the off-diagonal terms, and establish the explicit necessary and sufficient conditions for the zero solution to be asymptotically stable in terms of the coefficients and delay parameters.

Title : Geometric McKay Correspondence

Speaker : Haruko Matsuzawa (Osaka Metropolitan University)

Abstract : I will introduce what is so-called the geometric McKay correspondence. This is a correspondence between irreducible representations of a finite subgroup $G$ of $\it{SL}(2, \mathbb{C})$ and the exceptional divisors of the minimal resolution of the quotient singularity $C^2/G$. This phenomenon, relationship between representation theory and algebraic geometry, is based on the original McKay correspondence observed by John McKay in 1979, and it has been a subject of research from various perspectives up to the present day.

Title : A quandle which is related to the relative homotopy group of a knotted surface exterior with a handle decomposition

Speaker : Ryosuke Miki (The University of Osaka)

Abstract : It is well known a handle decomposition of the exterior of a knotted surface can be read from its banded link presentation. Using this fact, J.F. Martins established a method to compute the relative homotopy group of a knotted surface exterior. Moreover, it is well known that the quandle is a very powerful tool in knot theory. In this talk, I will introduce a quandle related to the relative homotopy group of a knotted surface exterior with a handle decomposition read from its banded link presentation, and introduce a quandle crossed module involving this quandle.

Title : Solvable Lie algebras obtained by quivers and Einstein metrics

Speaker : Fumika Mizoguchi (Osaka Metropolitan University)

Abstract : Nilpotent and solvable Lie groups are interesting subjects in the study of special left-invariant geometric structures on Lie groups. In our previous work, we constructed nilpotent Lie algebras from finite quivers without cycles by utilizing the concept of paths within quivers, and we proved that the corresponding simply-connected Lie groups admit Ricci solitons. In this talk, we extend our approach by constructing solvable Lie algebras from finite quivers without cycles, adding vertices as paths of length zero. We prove that if a quiver is an oriented multi-tree, the solvable Lie group obtained by this quiver admits a left-invariant metric, which is isometric to the direct product of a flat metric and an Einstein metric.

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Speaker : Junseok Moon (Kyungpook National University)

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Title : Mathematical Modelling of Malaria: Mother-to-Child Transmission and Dragonfly Predation

Speaker : Joy Nana Okogun-Odomley (Pusan National University)

Abstract : Malaria remains a major public health concern, particularly in Sub-Saharan Africa, where it causes significant morbidity and mortality. While traditional vector control methods such as insecticide-treated nets and indoor spraying have had positive impacts, the increasing resistance of mosquito populations to chemical interventions calls for sustainable and ecologically viable alternatives. This study proposes a compartmental model for malaria transmission that incorporates dragonfly predation as a natural mosquito control mechanism. The model extends classical SIR-type frameworks by including exposed compartments for both human and mosquito populations and introduces a dragonfly predation term into the mosquito dynamics. Dragonflies are effective predators of mosquitoes at both larval and adult stages, and their integration into the model represents a biologically grounded approach to reducing vector populations.

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Speaker : Sanghoon Park (Pusan National University)

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Speaker : Daesick Ryu (Kyungpook National University)

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Title : One-dimensional boundary blow-up problem with an exponential nonlinearity

Speaker : Kazuki Sato (Osaka Metropolitan University)

Abstract : "A boundary blow-up problem" refers to a boundary value problem in which the solution diverges to infinity near the boundary of the domain. It is known that such problems admit a solution when the nonlinear term satisfies the Keller–Osserman condition. In this talk, we consider a one-dimensional boundary blow-up problem with an exponential nonlinearity. We show that the problem admits a unique solution, which can be obtained in an explicit form. This is joint work with Futoshi Takahashi.

Title : Projective Dimensional Properties of the Hurwitz Module

Speaker : Galmandakh Tsog-Ochir (Kyungpook National University)

Abstract : Suppose $R$ be a commutative ring with identity, let $hR$ be the well-known Hurwitz polynomial ring over $R$ and let $M$ be an $R$-module. Then we construct the module over $hR$ as $HM=hR\otimes_R M$, which we call the Hurwitz module. In this talk, we compared Hurwtiz module with usual polynomial module ($R[x]$-module), and investigated under what condition the Hurwitz modules and the $R[x]$-modules would be isomorphic with each other. Furthermore, we explored some properties on projective dimension of the Hurwitz module $HM$.

Title : 2-dimensional braids over $P^2_-$ of degree two

Speaker : Haruchika Tsuno (The University of Osaka)

Abstract : A 2-dimensional braid over $P^2_-$ is an analogue of a closed 2-dimensional braid, introduced by O. Viro and S. Kamada. It is a branched covering surface embedded in a regular neighborhood of a standard projective plane $P^2_-$ in $\mathbf{R}^4$, regarded as a disk bundle over $P^2_-$. In this talk, we will introduce the chart presentation of 2-dimensional braids over $P^2_-$, a method to represent them by graphs, and classify 2-dimensional braids over $P^2_-$ of degree two by the chart presentation.

Title : On the Bott--Chern and Aeppli cohomology of $2$-dimensional theta toroidal groups.

Speaker : Jinichiro Tanaka (Osaka Metropolitan University)

Abstract : A toroidal group $X$ is a connected complex abelian Lie group without non-constant holomorphic functions, and can be regarded as a generalization of a complex torus. If all the Dolbeault cohomologies $H^{p,q}(X, \mathcal{O}_X)$ are finite-dimensional, it is called a theta toroidal group. Umeno identified the de Rham cohomology $H^k_{DR}(X,\mathbb{C})$, and demonstrated the Hodge decomposition $H^k_{DR}(X,\mathbb{C}) = \bigoplus_{p+q=k}H^{p,q}(X, \mathcal{O}_X)$ for every theta toroidal group $X$. Thus, theta toroidal groups resemble complex tori as compact complex manifolds. Then we focus on Bott--Chern and Aeppli cohomology, which play important roles in the study of compact complex manifolds. In this talk, we will observe the Bott--Chern and Aeppli cohomology of non-compact $2$-dimensional theta toroidal groups.

Title : Geometry of left-invariant pseudo-Riemannian metrics on Lie groups and closed orbit spaces

Speaker : Yuri Yamashita (Osaka Metropolitan University)

Abstract : In geometry, the classification of distinguished metrics is an important issue. In this research, we consider distiguished left-invariant pseudo-Riemannian metrics on Lie groups. It is a natural and important problem to determine whether a given Lie group G admits some distinguished left-invariant pseudo-Riemannian metrics or not. For a given Lie group, there is a natural group action on the space of left-invariant pseudo-Riemannian metrics by the scalar multiplication and automorphisms. The orbit space of this group action is called the moduli space. However, in general, the moduli space is not Hausdorff and is complex. Therefore, in this research, we particularly focus on the closed orbit space among these orbit spaces. In this talk, we introduce that it is sufficient to consider metrics only on the closed orbit space to study existence and nonexistance of distinguished metrics. Additionally, we have determined the closed orbit space for some Lie groups. This result suggests a relationship between distinguished points in the closed orbit space and distinguished metrics.

Title : On example of nef，big and non-semipositive line bundles

Speaker : Yangyang Zhang (Osaka Metropolitan University)

Abstract : Kodaira's embedding theorem states that if a compact $K\ddot{a}hler$ manifold $X$ admits a positive line bundle $L$, then $X$ can be embedded into projective space. In other words, $L$ is ample. From this perspective, the algebraic-geometric properties of being nef and big are considered generalizations of ampleness. Therefore, in version of Kodaira's theorem, it is natural to expect that such line bundles might also be semipositive. Based on this observation, I have conducted research starting from the question: Does there exist a line bundle that is nef and big but not semipositive? In the case of dimension one, it is known that every nef and big line bundle is positive. However, in what might be seen as a “surprising” result, Dano Kim constructed an explicit example in dimension three or higher of a line bundle that is nef and big but not semipositive. This naturally shifts attention to the two-dimensional case. Filip and Tosatti proposed the following conjecture: The line bundle in Grauert's example is expected to be nef and big but not semipositive. Based on this conjecture, Koike proved that the line bundle in a modified version of Grauert's example is indeed nef and big but not semipositive. The goal of this talk is to prove that Filip and Tosatti's conjecture is true by analyzing the structure of the original Grauert example and using the tool of the first obstruction class.