Winter School on Low-dimensional Topology and Related Topics

Program

* The schedule may vary.

Abstracts:

Mini-courses :

• Tara Brendle (University of Glasgow)
• · Braids and mapping class groups

The mapping class group MCG(S) of a surface S is the group of symmetries of S, that is, the group of self-homeomorphisms of S up to isotopy; braid groups are the special case where the surface S is a punctured disk. MCG(S) acts naturally on the first homology group of S, giving rise to a representation MCG(S) → Sp(2g,Z). The Torelli group of the surface S is the kernel of this representation, and is often described as the “nonlinear” or “mysterious” part of the mapping class group. A basic question about any group is: what is its abelianization, and what does that tell us about the group? In 1978, Birman-Craggs discovered the first abelian quotients of the Torelli group, giving a family of maps to Z/2 via the Rokhlin invariant of 4-manifolds. A few years later, Dennis Johnson discovered a new abelian quotient of the Torelli group onto a free abelian group. He further showed that this map, now known as the Johnson homomorphism, together with the Birman-Craggs maps, are sufficient to calculate the abelianization of the Torelli group.

In these talks, we will explain both “pieces” of the abelianization: we will describe the Birman-Craggs maps and also give two ways to define the Johnson homomorphism, one in terms of certain 3-manifolds and one that is more algebraic in flavor. We will also briefly survey more recent work of Masatoshi Sato and of Tudur Lewis on the abelianization of a closely related group, the level 2 congruence subgroup of MCG(S), and describe how this sheds new light on the work of Johnson and Birman-Craggs. Finally, we will describe aspects of Johnson theory in the context of braid groups.

• Zsuzsanna Dancso (The University of Sydney)
• · Algebraic Knot Theory

The notion of formality originates in the early rational homotopy theory of the 70's, and it has become a powerful notion in many other fields including group theory and Lie theory. In knot theory formality isomorphisms provide powerful invariants, studied under the synonymous names universal finite type/ Vassiliev invariants, universal quantum invariants or homomorphic expansions. The first prototype for a such an invariant is the Kontsevich integral: this was one of the results which won Kontsevich the Fields medal in 1998.

In this mini course we formalise (ha!) a method - a recipe, if you will - for constructing and studying formality isomorphisms for classes of knotted objects: most prominently knotted objects which can be finitely presented as a kind of algebraic structure. A prime example is the braid group, which has a famous finite presentation, the Artin presentation; other examples abound, from tangles viewed as a planar algebra to virtual/welded tangles viewed as a circuit algebra, tensor category or PROP, and even knotted graphs with their own strange and unique operations.

We'll investigate a range of examples of structure-preserving formality isomorphisms from the easy (welded braids) through the hard (knots, welded foams), to the impossible (tangles). We'll see how these invariants provide deep connections to various branches of algebra, again from simple (exponential maps) to complex (tensor categories, Drinfeld associators, Lie Theory, Kashiwara-Vergne and Duflo theory). Finally, we'll learn to use these connections to import tools and theorems between from topology to algebra and vice versa, and explore areas where this could be further exploited.

• Jun Murakami (Waseda University)
• · Volume conjecture of knots

In this series of talks, I would like to explain the volume conjecture, which gives a bridge between quantum invariants and geometric structures of knots. In 1980's, V. Jones discovered the Jones polynomial of knots. This invariant relates to the quantum group and generalized by various ways. In 1990's, R. Kashaev invented quantum invariants and observed that certain limit of his invariants converges to the hyperbolic volume of the knot complement. Kashaev's invariants turned out to be specializations of the colored Jones polynomial, and his observation led to the volume conjecture which predicts the relation between the colored Jones polynomial and the volume of the knot complement. I first introduce quantum invariants of knots and its relation to quantum groups. Then explain the volume conjecture. For some simple knots, proof is given and the idea of the proof is explained. Even though the volume conjecture is not solved yet for general case, the relation between colored Jones invariants and the hyperbolic structure of knot complements is known. This relation is a good clue that the conjecture actually holds. Some generalizations and applications of the volume conjecture are also explained.

• Jessica Purcell (Monash University)
• · Hyperbolic knot theory

It has been known since work of Thurston over fifty years ago that many knots (in an appropriate sense, “most” knots) have a complement that admit a hyperbolic structure. This structure is known to be unique up to isometry. Thus, hyperbolic geometry is a complete knot invariant. However, knots are classically presented by a diagram: a graph with over-under crossing information. It remains difficult to extract hyperbolic geometric information from the diagram of a knot, even if the knot is known to be hyperbolic. There have been many tools developed that give insight into hyperbolic geometry; these tools can be particularly effective for special classes of knots. In these talks, I will give background on hyperbolic geometry and present some of the techniques that have been leading to new results in knot theory. These will include triangulations and gluing equations, geometric deformation and Dehn filling, and tools used to bound hyperbolic volumes of knot complements. I will also point to a few open conjectures along the way.

Speakers, titles and abstracts :

• Adnan (Kangwon National University)
• · Alexander polynomial of twisted torus knots

Abstract: Twisted torus knots are a generalization of torus knots obtained by introducing additional full twists to adjacent strands of torus knots. In this talk, we present an explicit formula for the Alexander polynomial of twisted torus knots. We use a presentation of the knot group of twisted torus knots and Fox’s free differential calculus. We further explore the applications of our computations, including a determination of the genus for certain families of twisted torus knots. This is joint work with Kyungbae Park.

• Byunghee An (Kyungpook National University)
• · Quasi-isometry classification of graph-2-braid groups

Abstract: For a compact (weakly) special square complex, the intersection complex which is a certain complex-of-group decomposition structure of its fundamental group is a well-defined invariant under quasi-isometry. In this talk, we use this invariant to classify quasi-isometry types of 2-braid groups of circumference 1 graphs. As an application, we also classify quasi-isometry types of 4-braid groups of trees.

• Thiago de Paiva Souza (Monash University/IMPA)
• · Satellites and Lorenz knots

Abstract: Morton conjectured that every Lorenz knot that is a satellite is a cable on a Lorenz knot. In this talk, we construct infinitely many families of Lorenz knots that are satellites but not cables, giving counterexamples to this conjecture.

• Connie On Yu Hui (Monash University)
• · Modular links: Bunch algorithm and upper volume bounds

Abstract: In the 1970s, Williams developed an algorithm that has been used to construct and study modular links in the Lorenz template. We introduce the bunch algorithm to provide more insights into the geometry of modular links and Lorenz links. Using the machinery developed for the bunch algorithm, we provide the first upper volume bound that is independent of word exponents and quadratic in the braid index of the Lorenz link component for all modular link complements. We find families of modular knot complements with upper volume bounds that are linear in the braid index. A classification of modular link complements based on the relative magnitudes of word exponents is also presented.

• Gyo Taek Jin (KAIST)
• · Strong quasipositivity and arc index

Abstract: We present a viewpoint on Euler characteristic 0 braided surfaces as grid diagrams. This leads to some results regarding estimates of Thurston-Bennequin invariants of knots, strong quasipositivity of Whitehead doubles, jump numbers of slice-torus invariants, and arc and braid index.

• Naoki Kimura (Waseda University)
• · Some generalizations of racks and invariants of Legendrian knots

Abstract: Racks and quandles are algebraic systems which bring knot invariants. In this talk, we introduce algebraic systems called a bi-Legendrian rack and a 4-Legendrian rack, both of which are racks equipped with additional structures. We explain these algebraic systems provide invariants of Legendrian knots.

• Patrick Kinnear (University of Edinburgh)
• · An invertible non-semisimple TQFT varying over the character stack

Abstract: In this talk I will describe the construction of an invertible sheaf of vector spaces (i.e. a line bundle) on the G-character stack of a 3-manifold, which is the moduli stack of G-local systems for a fixed reductive group G. In fact, this is just one part of a TQFT which has a nice invertibility property relative to G-gauge theory, constructed from the representations of the quantum group at a root of unity. At the level of surfaces we obtain an invertible sheaf of categories over the character stack, which is expected to relate to a sheaf of algebras whose global sections are the skein algebra of the surface. Invertibility of the sheaf of categories should relate to an invertibility property of the skein algebra called being Azumaya. I will explain how this TQFT is expected to relate to other root-of-unity invariants such as those constructed by Akutsu-Deguchi-Ohtsuki and Costantino-Geer-Patureau-Mirand.

• Yuka Kotorii (Hiroshima university)
• · "Link-homotopy" of spatial graphs

Abstract: A link-homotopy of links is an equivalence relation on links generated by ambient isotopies and self-crossing changes, defined by Milnor. The self-crossing change is a crossing change between the same components. A spatial graph is an embedding of an abstract graph in $S^3$. A component-homotopy is a link-homotopy for spatial graphs, defined by Fleming. In this talk, we give a Markov type theorem for component-homotopy classes of spatial graphs, by using Habegger-Lin’s idea for link-homotopy classes of string links. This research is joint work with Atsuhiko Mizusawa.

• Erika Kuno (Osaka University)
• · Automorphisms of fine curve graphs for nonorientale surfaces

Abstract: The fine curve graph of a surface was introduced by Bowden, Hensel, and Webb as a graph consisting of the actual essential simple closed curves on the surface. Long, Margalit, Pham, Verberne, and Yao proved that the automorphism group of the fine curve graph of a closed orientable surface is isomorphic to the homeomorphism group of the surface. We generalized their result to closed nonorientable surfaces $N$ of genus $g \geq 4$. This is a joint work with Mitsuaki Kimura.

• Komal Negi (Indian Institute of Technology Ropar)
• · Twisted Virtual Braids and Twisted links

Abstract: Twisted knot theory introduced by M. Bourgoin is a generalization of knot theory. It leads us to the notion of twisted virtual braids. In this talk we show theorems for twisted links corresponding to the Alexander theorem and the Markov theorem in knot theory. We also provide a group presentation and a reduced group presentation of the twisted virtual braid group.

• Sunul Oh (POSTECH)
• · Hyperbolic Dehn filling, volume, and transcendentality

Abstract: Let $M$ be a 1-cusped hyperbolic 3-manifold. In this talk, we discuss experimental results concerning the behavior of the number $N_M(v)$ of Dehn fillings of $M$ with a given volume $v$. We also give necessary conditions for Dehn fillings that share the same complex volume or exhibit conjugate complex volume differences. Additionally, we show the transcendentality of the Neumann-Zagier volume formula, proving that the growth of $N_M$ is slower than any power of its filling coefficient.

• Yuko Ozawa (Meiji University)
• · Epimorphisms between genus two handlebody-knot groups

Abstract: For two genus $g$ handlebody-knots $H_{1}$ and $H_{2}$, we denote by $H_{1}\geq H_{2}$ if there exists an epimorphism from the (handlebody-)knot group of $H_{1}$ onto the one of $H_{2}$. In the case of $g=1$ (knot case), this ``$\geq$" is a partial order for prime knots and has been determined up to 11 crossings. In this talk, we consider the case of $g=2$ and exhibit a lot of ordered pairs of irreducible handlebody-knots in the table up to 6 crossings.

• Alexander Stoimenov (Dongguk University, WISE campus)
• · Burau representation, stabilization and exchange moves of braids

Abstract: We discuss how the Burau representation can be used to obstruct to reducibility and exchangeability of braids. We determine the exact location of Burau eigenvalues of reducible and exchangeable braids (for suitable parameter). We apply this to show non-conjugate irreducible representatives of many knots.

• Masaaki Suzuki (Meiji University)
• · Twisted Alexander Vanishing order

Abstract: Suppose that there exists a surjective homomorphism of a knot group onto a finite group. The composition with the regular representation of the finite group provides the twisted Alexander polynomial of the knot. We focus on the order of the smallest finite group so that the corresponding twisted Alexander polynomial is zero, which we call TAV order (Twisted Alexander Vanishing order). In this talk, we see some examples and properties of TAV order.

Posters:

• Adnan (Kangwon National University)
• · Alexander polynomial of twisted torus knots

Abstract: Twisted torus knots are a generalization of torus knots obtained by introducing additional full twists to adjacent strands of torus knots. In this talk, we present an explicit formula for the Alexander polynomial of twisted torus knots. We use a presentation of the knot group of twisted torus knots and Fox’s free differential calculus. We further explore the applications of our computations, including a determination of the genus for certain families of twisted torus knots. This is joint work with Kyungbae Park.

• Connie On Yu Hui (Monash University)
• · Modular links: Bunch algorithm and upper volume bounds

Abstract: In the 1970s, Williams developed an algorithm that has been used to construct and study modular links in the Lorenz template. We introduce the bunch algorithm to provide more insights into the geometry of modular links and Lorenz links. Using the machinery developed for the bunch algorithm, we provide the first upper volume bound that is independent of word exponents and quadratic in the braid index of the Lorenz link component for all modular link complements. We find families of modular knot complements with upper volume bounds that are linear in the braid index. A classification of modular link complements based on the relative magnitudes of word exponents is also presented.

• Patrick Kinnear (University of Edinburgh)
• · Skein module dimensions for mapping tori of the 2-torus.

Abstract: The (Kauffman bracket) skein module of a 3-manifold is a C(q)-vector space related to the representation theory of quantum SL_2. It is a 3-manifold invariant which generalises the Jones polynomial of a knot. It was recently shown that skein modules of closed, compact, oriented 3-manifolds are generically finite dimensional, however the proof is not constructive and one key research goal of quantum topologists is to describe these vector spaces explicitly. In this poster I will present my recent computations of the skein modules of mapping tori of T^2. This gives the skein dimension for a new family of 3-manifolds (to date, the skein module dimension has been computed for the 3-torus, the product of a surface with a circle, and some other families of 3-manifolds). My results show that skein theory cannot be extended to a 4d TQFT, and my approach suggests how techniques from higher algebra and geometric representation theory can be brought to bear on skein-theoretic problems.

• Yuka Kotorii (Hiroshima university)
• · “Link-homotopy" of spatial graphs

Abstract: A link-homotopy of links is an equivalence relation on links generated by ambient isotopies and self-crossing changes, defined by Milnor. The self-crossing change is a crossing change between the same components. A spatial graph is an embedding of an abstract graph in $S^3$. A component-homotopy is a link-homotopy for spatial graphs, defined by Fleming. In this talk, we give a Markov type theorem for component-homotopy classes of spatial graphs, by using Habegger-Lin’s idea for link-homotopy classes of string links. This research is joint work with Atsuhiko Mizusawa.

• Sangsoo Lee (Kyungpook National University)
• · An estimate of the triple point number of a surface-knot colored by the 7-dihedral quandle

Abstract: The triple point number is one of the elementary invariants of a surface-knot analogous to the crossing number of a classical knot. It can be defined as the minimal number of triple points over all possible diagrams of the surface-knot. Satoh and Shima suggested an upper bound of the triple point number of a surface-knot. In this talk, we investigate a lower bound of the triple point number of a surface-knot colored by the 7-dihedral quandle.

• Raquel Magalhães de Almeida Cruz (Université de Caen Normandie and University of São Paulo)
• · Ordering Pure Braid Groups on the Disk

Abstract: Given $D$ the closed unit disk, we present the Artin-Magnus ordering of the pure braid group of the disk, namely $PB_n(D)$. This ordering is invariant under both left and right multiplication. We note that, with this order, $PB_n(D)$ is order-isomorphic to the additive group of the rational numbers. Moreover, we explore the concept of Garside-positive braids and its relation with the Artin-Magnus ordering. Finally, we state that pure braid groups on closed orientable surfaces are bi-orderable.

• Komal Negi (Indian Institute of Technology Ropar)
• · Twisted Virtual Braids and Twisted links

Abstract: Twisted knot theory introduced by M. Bourgoin is a generalization of knot theory. It leads us to the notion of twisted virtual braids. In this talk we show theorems for twisted links corresponding to the Alexander theorem and the Markov theorem in knot theory. We also provide a group presentation and a reduced group presentation of the twisted virtual braid group.

• Jinseok Oh (Kyungpook National University)
• · A rank range of the set-theoretic Yang-Baxter cohomology of Alexander biquandles

Abstract: A homology theory of set-theoretic Yang-Baxter operators was established by J. S. Carter, M. Elhamdadi, and M. Saito. It was generalized for pre-Yang-Baxter operators independently by V. Lebbed and J. H. Przytycki. Biquandles, a generalization of quandles, are special cases of set-theoretic Yang-Baxter operators. In this talk, we determine tight upper and lower bounds of the Betti numbers for the set-theoretic Yang-Baxter cohomology of finite Alexander biquandles. This is joint work with Xiao Wang, Seung Yeop Yang, and Hongdae Yun.

• Sunul Oh (POSTECH)
• · Hyperbolic Dehn filling, volume, and transcendentality

Abstract: Let $M$ be a 1-cusped hyperbolic 3-manifold. In this paper, we investigate the behavior of the number $N_M(v)$ of Dehn fillings of M with a given volume $v$. Our study focuses on 236 manifolds with cusp shapes characterized by rational real parts. We establish necessary conditions for Dehn fillings that share the same complex volume or exhibit conjugate complex volume differences. Additionally, we demonstrate the transcendental nature of the Neumann-Zagier volume formula, proving that the growth of $N_M$ is slower than any power of its filling coefficient.

• Alexander Stoimenov (Dongguk University, WISE campus)
• · Burau representation, stabilization and exchange moves of braids

Abstract: We discuss how the Burau representation can be used to obstruct to reducibility and exchangeability of braids. We determine the exact location of Burau eigenvalues of reducible and exchangeable braids (for suitable parameter). We apply this to show non-conjugate irreducible representatives of many knots.

• Masaaki Suzuki (Meiji University)
• · 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 a linear representation of the knot group. If there exists a surjective homomorphism of 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 poster, we provide several formulas of the twisted Alexander polynomial of a knot associated to such representations.

• Hongdae Yun (Kyungpook National University)
• · Exploring geometric realizations of the extreme Khovanov homology of 2-bridge knots and links

Abstract: In the study of knots, we use special tools called invariants to tell different knots apart. One powerful invariant is the Jones polynomial. Khovanov homology, introduced by Mikhail Khovanov in the early 2000s, takes the Jones polynomial categorified, connecting knot theory and homological theory. Furthermore J. González-Meneses, P.M.G. Manchón, and M. Silvero, have shown that the (potential) extreme Khovanov homology of a link is related to the independence simplicial complex of a Lando graph associated with that link. We investigate the (real) geometric realization of extreme Khovanov homology, focusing on 2-bridge links and knots. To do this, we recall the connection between graph homology and Khovanov homology. From this, we explore the homotopy type of (real) extreme Khovanov homology for specific groups of links. This work is a collaboration with Jinseok Oh and Seung Yeop Yang.

List of Participants

Yong-Geun Oh (IBS-CGP, POSTECH)
Anderson Vera (IBS-CGP)
Tara Elise Brendle (University of Glasgow)
Jessica Purcell (Monash University)
Zsuzsanna Dancso (The University of Sydney)
Jun Murakami (Waseda University)
Alexander Stoimenov (Dongguk University, WISE)
Byunghee An (Kyungpook National University)
Connie On Yu Hui (Monash University)
Sunul Oh (POSTECH)
Masaaki Suzuki (Meiji University)
Komal Negi (Indian Institute of Technology Ropar)
Yuko Ozawa (Meiji University)
Gyo Taek Jin (KAIST)
Erika Kuno (Osaka University)
Naoki Kimura (Waseda University)
Patrick Kinnear (University of Edinburgh)
Adnan (Kangwon National University)
Yuka Kotorii (Hiroshima university)
Thiago de Paiva Souza (Monash University/IMPA)
Min Hoon Kim (Kyungpook National University)
BoGwang Jeon (POSTECH)
Hyunseok Do (POSTECH)
Seonmi Choi (Kyungpook National University)
Wenbo Liao (Chinese university of Hong Kong)
Inyoung Ryu (Texas A&M University)
JunHo Ji (Kanwon National University)
Raquel Magalhães de Almeida Cruz (Université de Caen Normandie and University of São Paulo)
Seonhwa Kim (University of Seoul)
Philip Choi (Seoul National University)
Seo Donggyun (Seoul National University)
Atsuko Katanaga (Shinshu University)
Dong-hyun Lee (Seoul National University)
Hyejung Kim (Jeonbuk National University)
Sunghwan Ko (Seoul National University)
Byeorhi Kim (POSTECH)
Hongdae Yun (Kyungpook National University)
jinseok Oh (Kyungpook National University)
Sangsoo Lee (Kyungpook National University)
Seung Yeop Yang (Kyungpook National University)
Sumayya Noor (Kyungpook National University)
Volker Genz (IBS-CGP)
Jaekwan Jeon (IBS-CGP)
Jongmyeong Kim (IBS-CGP)
Se-Goo Kim (Kyung Hee University)
Myeong-Sang Cho (POSTECH)
Seungook Yu (POSTECH)
Jaeyoung Choi (POSTECH)
Sam Bardwell-Evans (IBS-CGP)
Wonjun Chang (POSTECH)

Accommodation

We regret to say that we cannot support your travel and local expenses unless the conference promised to pay. For booking accommodation in Pohang, please contact the hotel directly referring the list below.

There is usually a limited number of rooms available, so please make a reservation as soon as you can.

Room Type	POSCO Int'l Center	Hotel Yeongildae	Apple Tree Hotel	Tour de Pohang (Woman's Safety Sohotel)
Double Room	77,000 KRW	110,000 KRW	45,000 KRW	43,000 ~ 48,000 KRW (1 person)
Twin Room	88,000 KRW	132,000 KRW	-	58,000 ~ 63,000 KRW (2 persons)
Breakfast (1 person)	13,200 KRW	Free	Free	Free
Distance from Venue	15 min. walk	15 min. drive	15 min. drive	15 min. drive
Contact	+82-54-279-8500 Email. kskim9916@naver.com	+82-54-280-8900	+82-54-241-1234	+82-0507-1397-1234

* The above rate is as of August, 2023 (VAT included).

** If you would like to stay at POSCO International Center, please fill out the form (Click here to download) and send it to the hotel before October 25, 2023. (Email. kskim9916@naver.com) There is a limited number of rooms available, so please make your reservation quickly.

*** The rate and condition may vary.

How to get to POSTECH

Visit here for information about how to get to Pohang.

For the location of the conference site, click here for POSTECH Campus Map.

Visa

Please visit here for more information.

Contact

sojung@ibs.re.kr

Close