Schedule

10:00 - 11:00: Rinat Kashaev (Geneva): Knot polynomials from braided Hopf algebras with automorphisms. 

11:00 - 11:30: Coffee break

11:30 - 12:30: Christine RS Lee (Texas State University): A topological model for the HOMFLY-PT polynomial 

12:30 - 14:00: Lunch

14:00 - 15:00: Jorge Becerra (Groningen): Universal quantum knot invariants

15:00 - 15:30: Coffee break

15:30 - 16:30: Dror Bar-Natan (Toronto): Knot Invariants from Finite Dimensional Integration

>16:30: Drinks

Location

The workshop will take place at the Linnaeusborg, room 5173.0165 , located in the Zernike Campus.

The easiest way to reach the campus from the Groningen train station is to take the bus number 15 from the Stationsplein, which departures every 5-10 minutes. Besides, bikes can be rented at the station and a ride to campus takes 20 minutes.

Abstracts

Dror Bar-Natan (Toronto): Knot Invariants from Finite Dimensional Integration

For the purpose of today, an "I-Type Knot Invariant" is a knot invariant computed from a knot diagram by integrating the exponential of a Lagrangian which is a sum over the features of that diagram (crossings, edges, faces) of locally defined quantities, over a product of finite dimensional spaces associated to those same features.

• Q. Are there any such things? A. Yes.

• Q. Are they any good? A. They are the strongest we know per CPU cycle, and are excellent in other ways too.

• Q. Didn't Witten do that back in 1988 with path integrals? A. No. His constructions are infinite dimensional and far from rigorous.

• Q. But integrals belong in analysis! A. Ours only use squeaky-clean algebra.

Further material can be found at drorbn.net/g24.

Jorge Becerra (Groningen): Universal quantum knot invariants

In this talk I will give an overview of the research carried out in my PhD thesis, which revolves around (quantum) knot invariants, monoidal categories and Hopf algebras. I will focus on two functorial tangle invariants: the Reshetikhin-Turaev invariant and the universal invariant, valued in monoidal categories that are constructed from ribbon Hopf algebras. In this talk, I will present a Hopf algebra ID very closely related to the quantum group U_h(sl_2) for which the universal invariant of an arbitrary knot can be effectively computed. I will also outline recent work relating certain truncation of this universal invariant with the 2-loop polynomial, a strong knot invariant that encodes a certain part of the Kontsevich invariant. If time allows I will also say some words about two projects that were inspired by these ideas: one about the non-strictification of monoidal categories and the other about constructing a combinatorial model for the PROP for bialgebras that generalises previous work by Pirashvili and Habiro.

Rinat Kashaev (Geneva): Knot polynomials from braided Hopf algebras with automorphisms. 

Given a braided Hopf algebra H endowed with an automorphism, one can construct an R-matrix over the underlying vector space of H. In the case of Nichols algebras, this leads to multivariable knot polynomials generalising those related to Borel parts of small quantum groups. This is a joint work with Stavros Garoufalidis.

Christine RS Lee (Texas State University): A topological model for the HOMFLY-PT polynomial 

A topological model for a knot invariant is a realization of the invariant as graded intersection pairings on coverings of configuration spaces. In this talk I will describe a topological model for the HOMFLY-PT polynomial. I plan to discuss the motivation from previous work by Lawrence and Bigelow giving topological models for the Jones and SL_n polynomials, and our construction, joint with Cristina Anghel, which uses a state sum formulation of the HOMFLY-PT polynomial to construct an intersection pairing on the configuration space of a Heegaard surface of the link.