Asymptotic homology of graph braid groups
by Byung Hee An (Kyungpook National University, Korea)
We give explicit formulas for the asymptotic Betti numbers of the unordered configuration spaces of an arbitrary finite graph over an arbitrary field.
A note on the asymptotic behavior of the twisted Alexander polynomials for some hyperbolic knots
by Airi Aso (OCAMI, Japan)
R. M. Kashaev conjectured that the asymptotics of the Kashaev invariant of hyperbolic links gives the hyperbolic volume of the link compliment. H. Murakami and J. Murakami extended Kashaev's conjecture (volume conjecture) and H. Murakami, J. Murakami, M. Okamoto, T. Takata, and Y. Yokota proposed the complexification of the volume conjecture. On the other hand, H. Goda gave a fomula of hyperbolic volume with twisted Alexander polynomials. Furthermore, J. Park conjectured a relation between the Reidemeister torsion and the complex volume. In this talk, we try to consider a complexification of Goda’s formula. To this end, we observe the asymptotics of the twisted Alexander polynomials of some hyperbolic knots.
Three amusing examples
by J. Scott Carter (University of South Alabama, United States)
I want to review the constructions of Boy's surface, Giller's surface (its orientation double cover), and the 2-twist spun trefoil as it is presented in Fox's quick trip through knot theory. These constructions will be quite detailed, and information about the self-intersection sets will be presented.
Remarks on algebraic structures related to skein relations
by Seonmi Choi (Kyungpook National University, Korea)
In 1987, the HOMFLY-PT polynomial is a 2-variable polynomial invariant for oriented links discovered by Hoste, Ocneanu, Millett, Freyd, Lickorish, Yetter, Przytycki and Traczyk. Przytycki and Traczyk introduced a new algebraic structure, called the Conway algebra, and constructed invariants of oriented links valued in Conway algebras. The HOMFLY polynomial can be obtained from the invariant. In 2012, they defined a Kauffman bracket magma and constructed an invariant of framed links. In this talk, we will define some generalizations of Conway algebras and Kauffman bracket magmas. These algebraic structures satisfy some conditions related to skein relations and the marked skein relation. We will construct invariants valued in them via marked graph diagrams.
A grading virtual knots and its applications
by Myeong-Ju Jeong (Korea Science Academy of KAIST, Korea)
Virtual knots are equivalence relation of virtual knot diagram under the virtual moves. We analyze the virtual moves and grade virtual knots by using the degrees of crossings involved in the virtual moves. We can see the structure of virtual knots and get invariants of virtual knots by using this grading.
Biquandle module invariants of oriendted surface-links in \mathbb{R}^4
by Yewon Joung (Pusan National University, Korea)
In this talk, we introduce invariants of oriented surface-links by enhancing the biquandle counting invariant using biquandle modules. We show that bead colorings of marked graph diagrams are preserved by Yoshikawa moves and hence define enhancements of the biquandle counting invariant for surface-links. This is joint work with Sam Nelson.
Pseudo Goeritz invariants of virtual link diagrams
by Naoko Kamada (Nagoya City University, Japan)
For a classical link diagram, a Goeritz matrix is defined by using a checkerboard coloring, However not every virtual link diagram admits a checkerboard coloring. We define the maps from the set of virtual link diagrams to that of almost classical virtual link diagrams. Almost classical virtual link diagrams admit checkerboard colorings. By use of these maps we introduce two kinds of matrices, which we call pseudo Goeritz matrices, for virtual link diagrams. Invariant factors of these matrices are invariants of virtual links. We show that one of them is also an invariant for a welded link.
Polynomial invariants of some family of knots
by Taizo Kanenobu (OCAMI, Japan)
We consider a certain family of knots, which share the same HOMFLYPT and the Q polynomial of Brandt, Lickorish, Millett, and Ho.
Infinite family of irreducible immersed 2-knots with one self-intersection point
by Kengo Kawamura (Kogakuin University, Japan)
Using a certain spinning construction producing immersed 2-knots, we give an infinite family of irreducible immersed 2-knots with one self-intersection point.
Smooth homotopy 4-sphere
by Akio Kawauchi (OCAMI, Japan)
It is explained how every smooth homotopy 4-sphere is diffeomorphic to the 4-sphere.
Turaev genera of adequate links obtained from 2-tangle replacement
by Byeorhi Kim (POSTECH, Korea)
Turaev genus of a link is the minimal genus of surfaces obtained by constructing a cobordism between all A and B Kauffman state of $D$ that has a saddle corresponding to each crossing and capping off the boundary components of the cobordism with disks among all possible diagram $D$ for the link. One of most important results corresponding to Turaev genus of a link is that a link is alternating if and only if its Turaev genus is trivial so that Turaev genus is a method to measure the distance of a link from being alternating. The class of adequate links is a generalization of the class of alternating links in the study of the minimal crossing number corresponding to the Tait’s conjecture. In this talk, we calculate the Turaev genus of a link obtained from an adequate link diagram by replacing a crossing of the diagram with an adequate 2-tangle diagram and also give some interesting examples about adequate links and adequate tangles. This is a joint work with Seongjeong Kim.
Symmetric biquandles and invariants for unoriented surface-links
by Jieon Kim (Pusan National University, Korea)
A biquandle is a non-empty set with two binary operations. When an invariant is constructed by using biquandles, orientations are needed. Biquandles are generalizations of quandles. For costructing invariants by using quandles, we need orientations. Kamada and Oshiro defined symmetric quandles, which will be used for constructing invariants for unoriented (surface-)links. In this talk, we define symmetric biquandles, which are generalizations of symmetric quandles. Also we construct invariants for unoriented (surface-)links by using symmetric biquandles.
On Links in $S_{g} \times S^{1}$ and classical knot invariant
by Seongjeong Kim (Bauman Moscow State Technical University / Moscow Institute of Physics and Technology, Russia)
A virtual knot, which is one of generalizations of knots in $\mathbb{R}^{3}$ (or $S^{3}$), is, roughly speaking, an embedded circle in thickened surface $S_{g} \times I$. In this talk we will discuss knots in $S_{g} \times S^{1}$, where $S_{g}$ is a surface of genus $g$. We introduce basic notions for knots in $S_{g} \times S^{1}$, for example, diagrams, moves for diagrams and so on. For knots in $S_{g} \times S^{1}$ technically we lose over/under information, but we will have information how many times the knot rotates along $S^{1}$. We will discuss the geometric meaning of the rotating information.
The Goeritz Matrices for general virtual knot
by Sera Kim (Republic of Korea Naval Academy, Korea)
Im, Lee and Lee announced the definition of the Goeritz matrix for checkerboard colorable virtual links in 2010. This is an extension of the Goeritz matrix for classical links in the $3$-dimensional Euclidean space. In this talk, we introduce the {\em incidence number} of odd crossings for a general virtual knot diagram and the definition of a generalized Goeritz matrix for virtual knots in thickened surfaces $S_g*[0,1]$. We also investigate some invariants from this matrices and show examples.
On the $\widetilde{H}$-cobordism group
by Dongsoo Lee (Seoul National University, Korea)
Kawauchi defined a group structure on the set of homology $S^1\times S^2$’s under an equivalence relation called $\widetidle{H}$-cobordism. This group receives a homomorphism from the knot concordance group, given by the operation of zero-surgery. In this talk, we show that the kernel of the zero-surgery homomorphism contains an infinite rank subgroup generated by topologically slice knots, and introduce related questions.
Arc presentations of Montesinos links
by Hwa Jeong Lee (Dongguk University-Gyeongju, Korea)
Let $L$ be a Montesinos link $M(-p,q,r)$ with positive rational numbers $p$, $q$, and $r$, each less than $1$, and $c(L)$ the minimal crossing number of $L$. In this talk, we construct arc presentations of $L$ with $c(L)$, $c(L)-1$, and $c(L)-2$ arcs under some conditions for $p$, $q$, and $r$. Furthermore, we determine the arc index of infinitely many Montesinos links.
Knots in the full torus: ubiquituous virtual knot theory
by Vassily O. Manturov (Moscow Institute of Physics and Technology, Russia)
Virtual knot theory has experienced various invariants unusual in topology; it suffices to mention picture-valued invariants, parity bracket, and homologywith pictorial gradings. These invariants allow to realise the principle "If a diagram is complicated enough then it realises itself" similarly to words in free groups with unique reduced representative for each element. To realise such invariants, one needs some initial topological information like non-trivial homology or homotopy. We discuss the realisation of this techniques for knots in the thickened torus and indicate further possible applications in geometry and topology. This result is partially based on joint work with Kim Seongjeong and Kim Sera.
Knots with Jones polynomials having Mahler measure $1$
by Maciej Mroczkowski (University of Gdańsk, Poland)
We construct four infinite families of prime knots with very special Jones polynomials: their coefficients are alternatingly $1$ and $-1$, starting and ending with $1$. This extends the well known examples of the Jones polynomials of the knots $4_1$ and $9_{42}$. Their spans are equal to $2n^2-2$ and $n^2+n-2$ for any $n\ge 2$. Such polynomials have Mahler measure equal to $1$, hence they are products of cyclotomic polynomials. In particular, we show that infinitely many roots of unity occur as zeros of Jones polynomials. On the other hand, we show that some inifinite families of roots of unity cannot occur as zeros of Jones polynomials of knots.
The intersection polynomials of a virtual knot
by Yasutaka Nakanishi (Kobe University, Japan)
We introduce three kinds of invariants of a virtual knot called the first, second, and third intersection polynomials. The definition is based on the intersection number of a pair of curves on a closed surface. We study several properties of the intersection polynomials and their applications.
Region Coloring Invariants
by Sam Nelson (Claremont McKenna College, United States)
In this talk we will see some recent results on region coloring structures in knot theory, including Niebrzydowski tribrackets and several generalizations.
On constituent links for handlebody-knots
by Shinya Okazaki (OCAMI, Japan)
A handlebody embedded in the $3$-sphere is called a handlebody-knot. When the genus $g$ handlebody-knot is cut open with $(g-1)$ meridian disks, a $g$-component knotted solid tori appears. This is regarded as a $g$-component link and is called a constituent link of the handlebody-knot. In this talk, we describe a necessary conditions for a $2$-component link to be a constituent link of a genus $2$ handlebody-knot in the table of links with up to nine crossings.
Computing extreme parts of Khovanov homology of 3- and 4-braids in polynomial time
by Józef H. Przytycki (The George Washington University, United States / University of Gdańsk, Poland)
It is proven by Morton and Short (1990) that computing the Jones and Homflypt polynomial of a closed braid of fixed number of strings has polynomial time complexity in the number of crossings. We showed in the joint work with T.Przytycka (1987/1993) that an essential part of the Jones-type polynomial link invariants can be computed in subexponential time. This is in a sharp contrast to the result of Jaeger, Vertigan and Welsh (1990) that computing the whole polynomial and most of its evaluations is $NP$-hard and is conjectured to be of exponential complexity. Our goal is to prove results generalizing that of Morton and Short for Khovanov homology. As a step in this direction we proved the following Theorem:
The time complexity of computing extreme Khovanov homology of closed 4-braids is polynomial and the homotopy type of geometric realization is either contractible or a sphere, or a wedge of two spheres.
Homotopy motions of surfaces in 3-manifolds
by Makoto Sakuma (OCAMI, Japan)
We introduce the concept of a homotopy motion of a subset in a manifold, and give a systematic study of homotopy motions of surfaces in closed orientable 3-manifolds. This notion arises from various natural problems in 3-manifold theory such as domination of manifold pairs, homotopical behaviour of simple loops on a Heegaard surface, and monodromies of virtual branched covering surface bundles associated to a Heegaard splitting. This is a joint work with Yuya Koda (arXiv:2011.05766).
From knot coloring to the algebraic theory of quandles
by David Stanovsky (Charles University in Prague, Czech Republic)
Quandle coloring provides an interesting knot invariant which is particularly useful for the algorithmic treatment of the knot equivalence problem. In the lecture, I will explain the algebraic tools that can be used to construct and enumerate various types of quandles suitable for knot coloring, including Joyce's classification of simple quandles, a group theoretic approach to connected quandles, and a universal algebraic approach to solvability and nilpotence for quandles.
An $f$-twisted Alexander matrix of quandles
by Yuta Taniguchi (Osaka University, Japan)
A quandle is an algebraic structure defined on a set with a binary operation whose definition was motivated from knot theory. Recently, A. Ishii and K. Oshiro introduced the $f$-twisted Alexander matrix, which is a quandle version of the twisted Alexander matrix. They showed that the twisted Alexander matrix can be recovered from the $f$-twisted Alexander matrix. In this talk, we study a relationship between $f$-twisted Alexander matrices and quandle cocycle invariants. As an application, we show that a square knot and a granny knot, which have the same knot group, are distinguished by an $f$-twisted Alexander matrix.
RightQuasigroups – a package for GAP with support for racks and quandles
by Petr Vojtechovsky (University of Denver, United States)
I will demonstrate a beta version of a package for GAP that supports one-sided quasigroups, such as racks and quandles. It allows creation of such objects in an intuitive way and it offers a number of computational methods based on permutation groups. The package contains extensive libraries of algebras. Since this is just a beta version, any input from the audience will be greatly appreciated.
Classification of 2-component virtual links up to forbidden detour moves
by Kodai Wada (Kobe University, Japan)
A forbidden detour move, independently introduced by Kanenobu and Nelson, is a local move on virtual link diagrams. This move is realized by the two forbidden moves in virtual knot theory. In particular, the equivalence relation on virtual links generated by forbidden detour moves implies the one by forbidden moves. This gives rise to the question: is the converse true? In 2021, Yoshiike and Ichihara proved that the question is true for virtual knots. In this talk, we show that it is true for 2-component virtual links by classifying them up to forbidden detour moves.
Yang-Baxter homology of cyclic biquandles
by Xiao Wang (Jilin University, China)
Solutions to the Yang-Baxter equation have shown its importance to the study of knot theory. Many link invariants have been built based on the homology theory of Yang-Baxter operators. In this talk, we will recall the definition of Yang-Baxter homology, give a basis for the free part of the Yang-Baxter homology of biquandles, and discuss an annihilation condition of its torsion part. This is a joint work with Seung Yeop Yang.
A normalization of A_2 bracket invariant for spatial graphs
by Tamotsu Basseda (Osaka City University, Japan)
A_2 bracket invariant defined by Kuperberg can be thought as a regular isotopy invariant for spatial graphs. I made it ambient isotopy invariant. I show how it can be normalized and that the ambient isotopy invariant has some natural properties.
Moduli Spaces and left-invariant symplectic structures on Lie groups
by Luis Pedro Castellanos Moscoso (Osaka City University, Japan)
We are interested in the classification or finding conditions for the existence of left-invariant symplectic structures on Lie groups. Some classifications are known, especially in low dimensions. We approach this problem by studying the "moduli space of left-invariant nondegenerate 2-forms", which is a certain orbit space in the set of all nondegenerate 2-forms on a Lie algebra. We present some of the results obtained so far with our approach, including a classification of left-invariant symplectic structures on some almost abelian Lie algebras.
Asymptotic behavior of least energy solutions to the Finsler Lane-Emden problem in dimension two
by Sadaf Habibi (Osaka City University, Japan)
We are concerned with the least energy solutions to the Lane-Emden problem driven by an anisotropic operator, so-called the Finsler Laplacian, on a bounded planar domain. We prove several asymptotic formulae as the nonlinear exponent gets large. This talk is based on an ongoing work with Prof. F. Takahashi (OCU).
Index polynomial invariants for twisted links
by Hiroki Ito (Osaka University, Japan)
Twisted links, which are extentions of virtual links, are defined by Bourgoin in 2006. In this talk, we introduce index polynomial invariants for ordered twisted links. For ordered virtual links, they coincide with affine index polynomial invariants defined by Kauffman in 2013.
Basic properties and various examples of weakly Arf rings
by Rihito Ito (Osaka City University, Japan)
A commutative ring is an algebraic system that has an addition, a subtraction, and a multiplication. Commutative algebra was started primarily to provide the basis for number theory and algebraic geometry. Inspired by fields such as combinatorics, computer algebra, topology, and invariant theory, it continues to develop. The main research subject of commutative algebra is the commutative Noetherian ring. There are various properties of the Noetherian ring. Among them, we talk about Arf and weakly Arf rings in this talk. Generalizing the Arf ring studied by J. Lipman, E. Celikbas, O. Celikbas, C. Ciuperca, N. Endo, S. Goto, R. Isobe, and Naoyuki Matsuoka defined a weakly Arf ring, and started studying its properties and examples. After a closer look, O. Zariski's conjecture about the relationship between strict closures and Arf rings has been completely resolved after half a century. Focusing on this, I would like to summarize what I learned about weakly Arf rings and strict closures.
Orthonormal function series formulae for inversion of the attenuated conical Radon transform with a fixed opening angle
by Kihyeon Jeon (Kyungpook National University, Korea)
Several types of conical Radon transforms have been studied since the introduction of the Compton camera. Several factors of a cone of integration can be considered as variables, for example, a vertex, a central axis, and an opening angle. In this paper, we study the conical Radon transform with a fixed central axis and opening angle. Furthermore, we consider the attenuation effect in the conical Radon transform because it allows us to obtain a high-quality reconstruction image. We construct a nonseparable Hilbert space and its maximal orthonormal set. This maximal orthonormal set comprises the eigenfunctions of the attenuated conical Radon transform, i.e. singular value decomposition (SVD). Finally, the inversion formula of the attenuated conical Radon transform is deduced from the SVD.
Pseudo knots and the Goeritz matrix
by Suhyeon Jeong (Pusan National University, Korea)
In 1933, Goeritz described how a quadratic form could be obtained from a regular projection of a knot, and showed that some of the algebraic invariants of this form are invariants of the knot.
Radial part of jump processes on manifolds
by Hirotaka Kai (Osaka City University, Japan)
A Lévy process is a stochastic process which has the independent stationary increments.Hunt constructed the Levy process on Lie groups and the homogeneous space.Applebaum constructed it on Riemannian manifolds using the orthonormal frame bundle. There are so many researches and results about Lévy processes on euclidean space.But there are few prior studies about the global behaviors of Lévy processes on Riemannian manifolds. The purpose of this lecture is to introduce the way to construct the Lévy process on Riemannian manifolds and the recent researches for the large time behaviors of it. The recurrence and transience are the fundamental properties of such behaviors. To determine which properties the stochastic process have is a classical subject. Ofcourse, necessary and sufficient conditions for the recurrence and transience of the euclidean Lévy process are well known. One of the most effective way to research the global behaviors of stochastic processes is to investigate the radial part of them. So,I will show you some estimates of radial part of the Lévy process on Riemannian manifolds and recent results.
Growth of the Frobenius twists of rings of prime characteristic
by Ryota Kobayashi (Osaka City University, Japan)
Let $ R $ be a commutative ring of prime characteristic $p$. The map $F: R\rightarrow R$ $ (x \mapsto x^p)$ that maps each element $x$ of $R$ to its p-th power is a ring homomorphism and is called the Frobenius map. This mapping peculiar to positive characteristic has been found to be very useful for studying commutative rings and algebraic varieties of positive characteristic. For example, E. Kunz proved that the Frobenius map $F$ of a Noetherian ring of prime characteristic is flat if and only if it is a regular ring. In this talk, we summarize some known ring-theoretic results for the invariants defined by Huneke and Leuschke, called F-signatures, defined using the asymptotic behavior of the indecomposable decompositions of the Frobenius twists of $R$.
Knots and quantum algorithms
by BongJu Kim (Pusan National University, Korea)
In this talk, we deal with relations between knots and quantum algorithms. We also anticipate what is likely to be computed by quantum algorithms in knot theory. Roughly speaking, a quantum algorithm is a probabilistic algorithm that can be expressed by a sequence of unitary operators on finite dimensional complex Hilbert spaces. According to the mathematical formulation of quantum mechanics, quantum mechanics is probabilisitc and it's physical symmetry formulated by unitary operators. For this reason, quantum algorithms can be performed on a physical device using quantum mechanical phenomena. On the other hand, a fast quantum algorithm approximating the Jones polynomials, called AJL algorithm, is known. We deal whit this algorithm in the simplest example.
Search for torsion in Khovanov homology of 2-cabling of torus knots
by Dong Han Kim (Kyungpook National University, Korea)
Khovanov homology is a richer oriented link invariant that categorifies the Jones polynomial. In this talk, we first review O. Viro’s combinatorial definition of (framed) Khovanov homology, and investigate torsion in Khovanov homology of cablings of torus knots.
The dichotomy for list switch signed h-colouring
by Hyobeen Kim (Kyungpook National University, Korea)
The switch homomorphism problem Switch$(H)$ for a signed graph $H$ is known to be polynomial time solvable if $H$ has an switch-core with at most two edges and is otherwise $NP$-complete. We present results towards a similar dichotomy classification for the list version of the problem.
Relative energy for Hamiltonian flows in the interacting gas dynamics with Lagrangian description
by Kiwoong Kwon (Kyungpook National University, Korea)
It is well known that in a system of hyperbolic conservation laws, even a smooth initial data generates a classical solution defined locally in time and it fails to be a solution after the maximal time interval. Although the existence of a weak solution of the system is established, the uniqueness of the solution cannot be guaranteed, that is, a situation where physical nonsense can occur. To sort out a suitable solution among them, an additional restriction called an entropy inequality must be imposed. In continuum physics, the Second Law of thermodynamics is essentially a statement of stability. Actually, the entropy inequality implies the stability of classical solutions of the system. In this talk, we consider an Euler equation generated by a potential energy, describing an interacting gas dynamics model in one dimension. The Lagrangian formulation of the system and the concept of a relative entropy energy will be introduced. With the variational structure of Hamiltonian and the corresponding relative energy, we obtain a more concise stability of solutions compared to that in Eulerian description of the system.
Survey on a Topological Data Analysis Toolkit : giotto-tda
by Jongkyu Lee (Pusan National University, Korea)
As a relatively new topic, topological data analysis (TDA), which uses tools from algebraic and combinatorial topology to extract features that capture the shape of data, has provided a wealth of new insights into the study of data with variety applications. Lots of research has been done to easily apply this method to practitioners, and among them, giotto-tda for Python as a kind of machine learning toolkits was recently developed. In this talk, I will introduce giotto-tda with some basic applications.
Linearization along Levi-flat hypersurface with a unit circle bundle structure
by Satoshi Ogawa (Osaka City University, Japan)
Let $X$ be a smooth complex surface. We say the local coordinate (resp. defining function) system of a submanifold (resp. hypersurface) in $X$ is linearizable if its transition maps are multiplication of unitary constant. Linearization problem along a submanifold (resp. hypersurface) in $X$ is how we denote a sufficient condition of its linearizablity. In this talk, let $Y$ be a compact complex curve unrelated to $X$ and $M$ be a hypersurface in $X$ which has a structure of a unit circle bundle over $Y$. I will consider linearization problem along $M$ and introduce the main method of proof, KAM theory.
Quandle homology of dihedral quandles and its automorphism groups
by Jinseok Oh (Kyungpook National University, Korea)
The free parts of rack and quandle homology were completely determined, but little is known about the corresponding torsion parts, although some details regarding the orders of torsion elements in rack homology have been derived. For example, it is known that the order of a finite quasigroup quandle annihilates the torsion subgroup of its rack homology. In this talk, we will discuss the torsion subgroups of homology of automorphism groups of dihedral quandles. This is joint work with Seung Yeop Yang.
Graph Invariants in Quandle Coloring Quivers
by Sanghoon Park (Pusan National University, Korea)
In 2018, K. Cho and S. Nelson constructed the quandle coloring quiver. As an enhancement of a coloring space, it is a link invariant. So graph invariants of the quiver are also link invariants. K. Cho and S. Nelson already suggest one of invariant -indegree quiver polynomial- that is obtained by computing degree centrality of the graph. In this talk, we review other graph invariants and compare them to distinguish the quivers.
Classification based on Topological Data Analysis
by Minju Seo (Pusan National University, Korea)
Topological Data Analysis (TDA) is an emergent field that aims to discover topological information hidden in a dataset. One such tool is persistent homology, which provides a multiscale description of the homological features within a dataset. A useful representation of this homological information is a persistence diagram. It contain fruitful information about the shape of data. In this talk, we consider TDA structure and compute persistence diagram for open datasets using python package.
When knot theory meets protein folding
by Mohd Ibrahim Sheikh (Pusan National University, Korea)
Knotted proteins have been found in all kingdoms of life. Knots in proteins are not only functionally important but are also helpful in enhancing the stability to protein chains. In this paper, we study the membrane proteins (which are notoriously difficult to study experimentally) and attempt to classify certain membrane protein families on the basis of knots contained in each family.
Finite β expansions
by Fumichika Takamizo (Osaka City University, Japan)
Let β>1 and α be a reciprocal of β. For nonnegative number x, we have a β-expansion of x by using β-transformation. Denote by Fin(β) be the set of x having finite β-expansion. β has the finiteness (positive finiteness) property if the set generated by integers (nonnegative integers) and α is a subset of Fin(β). These conditions are proposed by Frougny and Solomyak. In my talk, we define the odometer associtated with a β-numeration system and characterize the finiteness or positive finiteness properties by this odometer.
Parafermionic bases of standard modules
by Ryo Takenaka (Osaka City University, Japan)
We consider the complex simple Lie algebra of type $A$ with an automorphism of order 2. Now, the pair of the Lie algebra and the automorphism gives the twisted affine Lie algebra. In this talk, we consider the irreducible highest weight module called the standard module. By using a vertex algebra construction, we determine the basis of the standard module. As an application, we give the character formula of the parafermionic space which is defined as a quotient space of the standard module, and describe the relationship with the $W$-algebra.
Quandle coloring quivers for virtual links
by Ryotaro Ueda (Osaka University, Japan)
S. Nelson and K. Cho introduced the notion of a quandle coloring quiver, which is a quiver-valued classical link invariant. In this talk, we extend Nelson-Cho's invariant to virtual links using virtual quandle colorings. Properties similar to classical link's quandle coloring quiver studied by Yuta Taniguchi are discussed.
A plat form for surface-links
by Jumpei Yasuda (Osaka University, Japan)
It is known that every surface-link, which is a closed surface embedded in 4-space, can be described as the closure of a 2-dimensional braid providing it is orientable. In this talk, we introduce a new method of describing a surface-link using a braided surface, which we call a plat form. We prove that every surface-link, not necessarily orientable, can be described in a plat form. The plat index for a surface-link is defined. In classical knot theory, the plat index of a link coincides with the bridge index. The plat index of a surface-link we introduce here is an analogy of them.
Torsion in the set-theoretic Yang-Baxter homology of some finite biquandles
by Hongdae Yun (Kyungpook National University, Korea)
The Yang-Baxter equation was first appeared in theoretical physics. Afterwards, it has become important in knot theory, quantum groups, etc. as well. Biquandles are special solutions to the set-theoretic Yang-Baxter equation which can be used to construct knot invariants. A homology theory for the set-theoretic Yang-Baxter equation was introduced by Carter, Elhamdadi, and Saito. In this talk, we review the definition of set-theoretic Yang-Baxter homology and study set-theoretic Yang-Baxter homology groups of finite Alexander biquandles.