Conference
The target audience for this conference is faculty, graduate and advanced undergraduate students who are working in the field of Low-dimensional Topology. The seminars at the conference will be on current research in the field.
Abstracts
Alex Manchester (Rice University)
Title: Satellite Constructions and Concordance
Abstract: In this talk, we will give background for satellite constructions of links and concordance. Then, we will describe some results about the relationship between the two, in particular detailing how satellite constructions are a fruitful way to produce topologically slice knots. We will sketch some of the basic ingredients for proofs in this area, and finally survey some relationships between these ideas and the theory of topological 4-manifolds.
Balarka Sen (TIFR)
Title: h-principle for stratified spaces
Abstract: Smale's sphere eversion paradox states that any two immersions of the 2-sphere inside the Euclidean 3-space are isotopic through immersions. This is an instance of a much more general phenomenon called $h$-principles, introduced by Gromov, wherein sufficiently overdetermined partial differential relations occurring in geometry of manifolds can be solved purely algebro-topologically. We shall discuss a generalization of $h$-principles in the setting of singular spaces, and prove as a corollary a version of the Smale-Hirsch theorem for positive-codimension immersions between stratified spaces. The proof involves algebraic topology pertaining to certain Space-valued variants of constructible sheaves over stratified spaces that we call ``stratified continuous sheaves". Time permitting, I will mention parallels between our results and the main theorem of stratified Morse theory by Goresky and MacPherson. This is a joint work with Mahan Mj.
Marc Kegel (Humboldt University Berlin)
Title: Characterizing and non-characterizing knots by 3-manifolds
Abstract: From a knot K, we can build 3-manifolds by performing Dehn surgery on that knot. We will discuss some new results explaining in which sense the diffeomorphism types of these 3-manifolds characterize the isotopy class of the knot K. This talk is based on joint work with Baker, Baker-McCoy, and Weiss.
Rama Mishra (IISER Pune)
Title: Understanding State sum models for Colored Jones polynnomial
Abstract: An nth Colored Jones polynomial is a generalization of the Jones polynomial. It is an important quantum invariant for knots and links. It is abstractly defined as a quantum trace of representations of Artin's braid group into the Endomorphisms of the tensor product of (n+1) dimensional representation of the quantum group U_q(sl(2,C)). Thus n=1 corresponds to the Jones polynomial. It is difficult to compute this invariant using definition. Certain State sum models are proposed to compute these polynomials. These are defined with the help of associating a graph to a given link diagram. They make the computation easy but lack in explaining any geometric idea behind them. In this talk we compare two important state sum models for colored Jones polynomial and provide a geometric understanding and show that even though they are defined using two different graphs, they are essentially the same.
Rima Chatterjee (University of Cologne)
Title: Knots in contact manifolds
Abstract: A contact manifold is a smooth manifold equipped with an extra geometric structure. Knots in contact 3-manifolds are interesting objects to study. I'll give a brief survey on interesting problems on contact topology and then discuss some of the classification problems of Legendrian knots (A Legendrian knot is everywhere tangent to the contact structure). Part of this talk will be based on joint work with Geiges-Onaran and with Etnyre-Min-Mukherjee. No background knowledge of contact topology will be assumed.
Shelly Harvey (Rice University)
Title: Ribbon obstructions for knots from the Jones nullity
Abstract: A slice knot is a knot which bounds a smoothly embedded disk in B^4. A ribbon knot is a special slice knot where the disk has no maxima with respect to the height function in B^4. One of the most interesting open questions in knot theory is whether every slice knot is ribbon, the so called Ribbon-Slice conjecture. In 2008, Eisermann showed that the Jones polynomial of an n-component ribbon link is divisible by the Jones polynomial of the n-component trivial link. Unfortunately, the Jones polynomial of the unknot is 1 so this doesn't give a ribbon obstruction for knots, only links with at least two components. We discuss ways of potentially getting a ribbon obstruction using satellite operators and some operators that don't work. This is work in progress.
Tanushree Shah (Alfred Renyi Institute of Mathematics)
Title: Torsion in tight contact manifolds
Abstract: Contact structures come in two flavors: tight and overtwisted. The classification of overtwisted contact structures is well understood as opposed to tight ones. If a 3-manifold has an incompressible torus then there are infinitely many tight contact structures on it. I will give a distinction for when is it easy to work with positive torsion and in which cases one cannot except a reasonable classification. If time permits I would like to give a glimpse of some new tools that are being developed which could be helpful to study torsion.
Tejas Kalelkar (IISER Pune)
Title: Algorithms to recognise knots
Abstract: A central question in knot theory is to recognise knots from their diagrams. I will begin with a survey of algorithms for knot recognition and then focus on a specific algorithm for hyperbolic knots which uses Pachner moves. A Pachner move is a local combinatorial change to the triangulation of a manifold. Any two geometric ideal triangulations of a cusped complete hyperbolic 3-manifold are related by a sequence of Pachner moves through topological triangulations. We give a bound on the length of this sequence in terms of a lower bound on the dihedral angles of the geometric triangulations. This leads to an effective algorithm to check the equivalence of geometrically triangulated hyperbolic manifolds and therefore of hyperbolic knots. This is joint work with Sriram Raghunath.