Abstract: The quantum analogue of an integer, also called a Gaussian q-integer, is a polynomial in an indeterminate 'q' specialising to the integer when q=1. These q-integers can be thought of as a refined version of integers, and appear in many areas of mathematics. In particular, in the theory of Reshetikhin-Turaev quantum link invariants, to each positive integer n corresponds an n-invariant such that the unknot has invariant the q-analogue of n. 

 

In this talk, we will use the generalisation of q-integers to all real numbers defined in 2020 by Morier-Genoud and Ovsienko to introduce an x-invariant for each real number x, interpolating the usual n-invariants. We will see that these x-invariants can be categorified using the language of webs, and are recovered by a well-designed specialisation of the HOMFLY-PT polynomial. We will explain in what sense our x-invariants are the unique reasonable interpolation of n-invariants. 

 

This talk is based on a joint work with Hoel Queffelec.