Winter School on Lowdimensional Topology and Related Topics
December 11–15, 2023
Pohang
Organizers
 YongGeun Oh (IBS Center for Geometry and Physics, POSTECH)
 Anderson Vera (IBS Center for Geometry and Physics)
Minicourses
Braids and mapping class groups
Algebraic Knot Theory
Volume conjecture of knots
Hyperbolic knot theory
Venue
 IBS POSTECH Bldg. #301
Registration
Click here for registration.
* Online streaming is not available
For contributed talks and posters, please register before 20231025
General registration before 20231120
Program
Time  Dec. 11 (Mon)  Dec. 12 (Tue)  Dec. 13 (Wed)  Dec. 14 (Thu)  Dec. 15 (Fri) 

8:00 – 8:30  Welcome and Registration 

8:30 – 9:00  Breakfast / Coffee  
9:00 – 10:15  Minicourse A Jun Murakami 
Minicourse C Jessica Purcell 
Minicourse D Tara Brendle 
Minicourse B Zsuzsanna Dancso 
Minicourse C Jessica Purcell 
10:15 – 10:30  Coffefe Break  
10:30 – 11:45  Minicourse B Zsuzsanna Dancso 
Minicourse A Jun Murakami 
Minicourse C Jessica Purcell 
Minicourse A Jun Murakami 
Minicourse D Tara Brendle 
11:45 – 11:50  Short Break  
11:50 – 12:25  Sunul Oh  Thiago de Paiva Souza  Connie On Yu Hui  Yuka Kotorii  Masaaki Suzuki 
12:25 – 14:00  Lunch  Group photo /Lunch 
Lunch  Closing remarks /Lunch 

14:00 – 14:35  Byunghee An  Minicourse B Zsuzsanna Dancso (14:00  15:15) 
Free  Minicourse B Zsuzsanna Dancso (14:00  15:15) 

14:35 –15:10  Naoki Kimura  
15:10 –15:30  Coffee break  Coffee break  
15:30 – 16:05  Patrick Kinnear  Alexander Stoimenov  Erika Kuno  
16:05 – 16:40  Coffee break //Poster session 
Yuko Ozawa  Gyo Taek Jin  
16:40 – 17:00  Poster Session  Coffee break  Coffee break  
17:00 – 17:35  Shahab Adnan  Komal Negi *  
17:35 – 18:00  
18:00 – 20:00  Dinner 
* The schedule may vary.
Abstracts:
Minicourses :
 Tara Brendle (University of Glasgow)
 · Braids and mapping class groups
 Zsuzsanna Dancso (The University of Sydney)
 · Algebraic Knot Theory
 Jun Murakami (Waseda University)
 · Volume conjecture of knots
 Jessica Purcell (Monash University)
 · Hyperbolic knot theory
The mapping class group MCG(S) of a surface S is the group of symmetries of S, that is, the group of selfhomeomorphisms 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, BirmanCraggs discovered the first abelian quotients of the Torelli group, giving a family of maps to Z/2 via the Rokhlin invariant of 4manifolds. 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 BirmanCraggs 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 BirmanCraggs maps and also give two ways to define the Johnson homomorphism, one in terms of certain 3manifolds 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 BirmanCraggs. Finally, we will describe aspects of Johnson theory in the context of braid groups.
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 structurepreserving 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, KashiwaraVergne 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.
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.
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 overunder 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 :
 Shahab Adnan (Kangwon National University)
 · Alexander polynomial of twisted torus knots
 Byunghee An (Kyungpook National University)
 · Quasiisometry classification of graph2braid groups
 Thiago de Paiva Souza (Monash University/IMPA)
 · Satellites and Lorenz knots
 Connie On Yu Hui (Monash University)
 · Modular links: Bunch algorithm and upper volume bounds
 Gyo Taek Jin (KAIST)
 · Strong quasipositivity and arc index
 Naoki Kimura (Waseda University)
 · Some generalizations of racks and invariants of Legendrian knots
 Patrick Kinnear (University of Edinburgh)
 · An invertible nonsemisimple TQFT varying over the character stack
 Yuka Kotorii (Hiroshima university)
 · "Linkhomotopy" of spatial graphs
 Erika Kuno (Osaka University)
 · Automorphisms of fine curve graphs for nonorientale surfaces
 Komal Negi (to confirm) (Indian Institute of Technology Ropar)
 · Twisted Virtual Braids and Twisted links
 Sunul Oh (POSTECH)
 · Hyperbolic Dehn filling, volume, and transcendentality
 Yuko Ozawa (Meiji University)
 · Epimorphisms between genus two handlebodyknot groups
 Alexander Stoimenov (Dongguk University, WISE campus)
 · Burau representation, stabilization and exchange moves of braids
 Masaaki Suzuki (Meiji University)
 · Twisted Alexander Vanishing order
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.
Abstract: For a compact (weakly) special square complex, the intersection complex which is a certain complexofgroup decomposition structure of its fundamental group is a welldefined invariant under quasiisometry. In this talk, we use this invariant to classify quasiisometry types of 2braid groups of circumference 1 graphs. As an application, we also classify quasiisometry types of 4braid groups of trees.
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.
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.
Abstract: We present a viewpoint on Euler characteristic 0 braided surfaces as grid diagrams. This leads to some results regarding estimates of ThurstonBennequin invariants of knots, strong quasipositivity of Whitehead doubles, jump numbers of slicetorus invariants, and arc and braid index.
Abstract: Racks and quandles are algebraic systems which bring knot invariants. In this talk, we introduce algebraic systems called a biLegendrian rack and a 4Legendrian rack, both of which are racks equipped with additional structures. We explain these algebraic systems provide invariants of Legendrian knots.
Abstract: In this talk I will describe the construction of an invertible sheaf of vector spaces (i.e. a line bundle) on the Gcharacter stack of a 3manifold, which is the moduli stack of Glocal 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 Ggauge 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 rootofunity invariants such as those constructed by AkutsuDeguchiOhtsuki and CostantinoGeerPatureauMirand.
Abstract: Milnor defined a concept of linkhomotopy of links, which is generated by ambient isotopies and selfcrossing changes. The selfcrossing change is a crossing change between the same components. A spatial graph is an embedding of an abstract graph in S3. A componenthomotopy is a linkhomotopy for spatial graphs, defined by Fleming. In this talk, we give a classification of componenthomotopy of spacial graphs, by using Habeggerlin’s idea for the linkhomotopy classification of string links.
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.
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.
Abstract: Let $M$ be a 1cusped hyperbolic 3manifold. 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 NeumannZagier volume formula, proving that the growth of $N_M$ is slower than any power of its filling coefficient.
Abstract: For two genus $g$ handlebodyknots $H_{1}$ 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 handlebodyknots in the table up to 6 crossings.
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 nonconjugate irreducible representatives of many knots.
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:
 Connie On Yu Hui (Monash University)
 · Modular links: Bunch algorithm and upper volume bounds
 Patrick Kinnear (University of Edinburgh)
 · An invertible nonsemisimple TQFT varying over the character stack
 Sangsoo Lee (Kyungpook National University)
 · An estimate of the triple point number of a surfaceknot colored by the 7dihedral quandle
 Raquel Magalhães de Almeida Cruz (Université de Caen Normandie and University of São Paulo)
 · Ordering Pure Braid Groups on the Disk
 Komal Negi (to confirm) (Indian Institute of Technology Ropar)
 · Twisted Virtual Braids and Twisted links
 Jinseok Oh (Kyungpook National University)
 · A rank range of the settheoretic YangBaxter cohomology of Alexander biquandles
 Sunul Oh (POSTECH)
 · Hyperbolic Dehn filling, volume, and transcendentality
 Alexander Stoimenov (Dongguk University, WISE campus)
 · Burau representation, stabilization and exchange moves of braids
 Masaaki Suzuki (Meiji University)
 · Twisted Alexander polynomials of knots associated to the regular representations of finite groups
 Hongdae Yun (Kyungpook National University)
 · Exploring geometric realizations of the extreme Khovanov homology of 2bridge knots and links
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.
Abstract: In this talk I will describe the construction of an invertible sheaf of vector spaces (i.e. a line bundle) on the Gcharacter stack of a 3manifold, which is the moduli stack of Glocal 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 Ggauge 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 rootofunity invariants such as those constructed by AkutsuDeguchiOhtsuki and CostantinoGeerPatureauMirand.
Abstract: The triple point number is one of the elementary invariants of a surfaceknot 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 surfaceknot. Satoh and Shima suggested an upper bound of the triple point number of a surfaceknot. In this talk, we investigate a lower bound of the triple point number of a surfaceknot colored by the 7dihedral quandle.
Abstract: Given $D$ the closed unit disk, we present the ArtinMagnus 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 orderisomorphic to the additive group of the rational numbers. Moreover, we explore the concept of Garsidepositive braids and its relation with the ArtinMagnus ordering. Finally, we state that pure braid groups on closed orientable surfaces are biorderable.
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.
Abstract: A homology theory of settheoretic YangBaxter operators was established by J. S. Carter, M. Elhamdadi, and M. Saito. It was generalized for preYangBaxter operators independently by V. Lebbed and J. H. Przytycki. Biquandles, a generalization of quandles, are special cases of settheoretic YangBaxter operators. In this talk, we determine tight upper and lower bounds of the Betti numbers for the settheoretic YangBaxter cohomology of finite Alexander biquandles. This is joint work with Xiao Wang, Seung Yeop Yang, and Hongdae Yun.
Abstract: Let $M$ be a 1cusped hyperbolic 3manifold. 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 NeumannZagier volume formula, proving that the growth of $N_M$ is slower than any power of its filling coefficient.
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 nonconjugate irreducible representatives of many knots.
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.
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. GonzálezMeneses, 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 2bridge 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
To be announced
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  +82542798500 Email. kskim9916@naver.com 
+82542808900  +82542411234  +82050713971234 
* 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.
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