Geometry Seminar

Date: April 16, 2018

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Christine Lee, UT Austin

Title: A knot with no tail

Abstract: In this talk, we will discuss the stability behavior of the U_q(sl(2))-colored Jones polynomial, a quantum link invariant that assigns to a link K in S^3 a sequence of Laurent polynomials {J_K^n(q)} from n=2 to infinity. The colored Jones polynomial is said to have a tail if there is a power series whose coefficients encode the asymptotic behavior of the coefficients of J_K^n(q) for large n. Since Armond and Garoufalidis-Le proved the existence of a tail for the colored Jones polynomial of an adequate knot, first conjectured by Dasbach-Lin, it has been conjectured that multiple tails exist for all knots. Moreover, the stable coefficients of the tail have been shown to relate to the topology and the geometry of the alternating link complement, prompting the Coarse Volume Conjecture by Futer-Kalfagianni-Purcell. I will talk about an unexpected example of a knot, recently discovered in joint work with Roland van der Veen, where the colored Jones polynomial does not admit a tail, and discuss potential ways to view this example in the context of the categorification of the polynomial, the aforementioned Coarse Volume Conjecture, and a general conjecture made by Garoufalidis-Vuong concerning the stability of the colored Jones polynomial colored by irreducible representations of Lie algebras different from U_q(sl(2)).