Workshop on Volume Conjecture and related topics in Knot Theory

Workshop on Volume Conjecture and related topics in Knot Theory

IISER Pune

Dec 17-21, 2018

Title and Abstracts

Schedule of Talks

All talks will be held in Madhava Hall on third floor of Main Building

Day: Monday Dec 17, Tuesday Dec 18, Wednesday Dec 19
Name: Abhijit Champanerkar, CUNY Title: Hyperbolic knot theory Abstract: In this series of three talks we will introduce ideas, tools and examples in hyperbolic knot theory. In the first talk we will review basic hyperbolic geometry in dimensions two and three, important structure theorems for hyperbolic 3-manifolds and ideal tetrahedra which are building blocks for hyperbolic 3-manifolds. In the second talk we will discuss ideal triangulations, gluing equations, Thurston's Dehn surgery Theorem and hyperbolic volume. In the third talk we will discuss explicit examples of hyperbolic knots and links, angle structures and polyhedral decompositions of knot and link complements. We will also see computational tools like SnapPy and snap to study and compute geometric invariants of hyperbolic knots and 3-manifolds.
Day: Tuesday Dec 18, Wednesday Dec 19
Name: Stefan Friedl, University of Regensburg Title: An introduction to twisted Alexander polynomials Abstract: Twisted Alexander polynomials are a generalization of the classical Alexander polynomial. We will give the definition of twisted Alexander polynomials and we will discuss several of the key properties. We will conclude with some open questions. We will only assume basic knowledge of topology (i.e. fundamental groups).
Day: Thursday Dec 20, Friday Dec 21
Name: Shashank Kanade, University of Denver Title: Rogers-Ramanujan-type identities and asymptotics Abstract: Rogers-Ramanujan identities are a fascinating pair of identities involving integer partitions. Besides being inherently interesting from the point of view of number theory and combinatorics, they have important connections to knot theory, representation theory, conformal field theory etc. I plan to cover a selection of following topics. (1) Basics of Rogers-Ramanujan identities, their many analogues, generalizations and some recent conjectures. (2) Asymptotics of Rogers-Ramanujan-type q-series and their relations to Nahm's conjecture. (3) Relations of Rogers-Ramanujan identities to representation theory. Parts of the talk will be based on joint works with Matthew C. Russell.
Day: Monday Dec 17, Tuesday Dec 18, Wednesday Dec 19
Name: Ilya Kofman, CUNY Title: An introduction to the colored Jones polynomial Abstract: These three lectures will prepare workshop participants for more advanced talks that involve the colored Jones polynomial. In the first lecture, we discuss the Jones polynomial from the perspectives of the Kauffman bracket and skein theory, and via representations of braid groups. In the second lecture, we present the colored Jones polynomial using skein algebras and Jones-Wenzl idempotents. In the third lecture, we present the colored Jones polynomial using quantum groups and R-matrices.
Day: Thursday Dec 20
Name: Rama Mishra, IISER Pune Title: State Sum models for Colored Jones Polynomial Abstract: Finding a closed formula for computing $J_N(K)$, the Colored Jones polynomial is a challenging problem. In this talk I will discuss some known “State Sum Models” for computing $J_N(K)$ and do the calculation for “Figure eight knot”.
Day: Wednesday Dec 19, Thursday Dec 20, Friday Dec 21
Name: Hitoshi Murakami, Tohoku University Title: An Introduction to the Volume Conjecture and its generalizations, I, II, and III Abstract: The volume conjecture states that the asymptotic behavior of the N-dimensional colored Jones polynomial of a knot evaluated at the Nth root of unity would give the volume of the knot complement. In this series of talks I will give an introduction to the conjecture and its generalizations. In the first talk, I will give a quick definition of the colored Jones polynomial and show a proof of the volume conjecture for the figure-eight knot. I will also show more examples of the colored Jones polynomials including two-bridged torus knots. In the second talk, I will tell a rough idea of the potential proof of the conjecture in the case of a hyperbolic knot. I will quickly review the hyperbolic geometry of three-dimension. Then I will describe how a triangulation of the knot complement is related to the colored Jones polynomial. When such a triangulation determines the complete hyperbolic structure, the volume would give the leading term of the asymptotic expansion of the colored Jones polynomial for a large N. In the third talk, I will describe generalizations, a refinement and a parametrization, of the volume conjecture. In the former generalization, we conjecture that the asymptotic behavior of the colored Jones polynomial would also contain information about the Chern-Simons invariant (the imaginary part of the complex volume) and the Reidemeister torsion. In the latter, if perturb the Nth root of unity slightly, then we would have information of the volume, the Chern-Simons invariant, and the Reidemeister torsion associated with a deformed, incomplete hyperbolic structure of a knot complement.
Day: Monday Dec 17, Tuesday Dec 18
Name: Speaker: Visakh Narayanan, IISER Pune Title: "Ribbon categories": invariant generating machines! I & II Abstract: (Part 1) It is a remarkable fact that a "collection" of realisations of a particular kind of a mathematical structure (a category), can also have a mathematical structure defined over it. The theme of this talk will be to describe what is called a "ribbon structure" on a category and to see how it is tied up with the theory of knots and links. The category of "framed tangles" is in a sense the most "free" example of a ribbon category. Given any other ribbon category and a choice of an object in it, there is a unique functor arising from this data called the "Reshetikhin-Turaev functor" and it directly descends to an invariant of framed knots and links. So a ribbon category is in some sense an "invariant generating machine" and it is desirable to have realisations of this structure. (Part 2) Universal enveloping algebra of a Lie algebra naturally can be made into a Hopf algebra. The category of finite dimensional representations of a "braided ribbon hopf algebra" can be shown admit a ribbon structure. But the universal enveloping algebra has a trivial ribbon structure on the category of its representations. But this algebra can be "quantized" to deform some structure while keeping the representation theory similar (and in fact more interesting). Magically the category of finite dimensional representations of these "quantized enveloping algebras" posses an excellent ribbon structure. And thus we obtain an infinite family of computable (quantum) invariants. And in the special case of quantized sl_2 the invariants obtained from its standard 2-dimensional representation coincides with the classical "Jones polynomial". This talk will also reveal one manifestation of the harmony between Mathematics and Physics.
Day: Tuesday Dec 18, Wednesday Dec 19, Thursday Dec 20
Name: Kate Petersen, Florida State University Title: Characters, Representations and the A-polynomial Abstract: Let M be a finite volume hyperbolic 3-manifold. The character variety of M is an algebraic set that encodes a lot of data about the manifold M itself. It is related to the set of geometric structures on M, and can be used to get data about interesting surfaces in M. A lot of this data can be packaged into a 2-variable polynomial, called the A-polynomial. I’ll introduce the character variety and the representation variety associated to a 3-manifold. I’ll provide some examples, and discuss the connection between these sets and the geometry of ideal triangulations of these manifolds. I’ll discuss the geometry of character varieties, and talk about some of the data they encode. Also, I’ll discuss the construction of the A-polynomial and its utility. If time permits, I will discuss related arithmetic invariants.
Day: Monday Dec 17
Name: P. Ramadevi, IIT Mumbai Title: Twist Knot Invariants and Volume Conjecture Abstract: In this talk, I will briefly discuss our results on twist knot invariants. Then describe our works on super-A-polynomials and generalised volume conjecture.
Day: Friday Dec 21
Name: Mahendar Singh, IISER Mohali Title: Knot invariants from quandles. Abstract: In recent years, quandles and their analogues have been used to construct many invariants for knots. We will give a quick review of these objects and some of the invariants. In particular, we will discuss some knot invariants and new properties of knot quandles that we obtained recently.

Organizers:

Rama Mishra (IISER Pune),

Abhijit Champanerkar (CUNY)

Tejas Kalelkar (IISER Pune)

Contact: ramamishra64 at gmail dot com