In the last dozen years, topological methods have been shown to produce a new pathway to study arithmetic statistics over function fields, most notably in Ellenberg-Venkatesh-Westerland's work on the Cohen-Lenstra conjecture. More recently, Ellenberg, Tran and Westerland proved the upper bound in Malle's conjecture over function fields by studying the twisted homology of configuration spaces. In this talk, we will give an overview of their framework and extend their techniques to study other questions in arithmetic statistics. As an example, we will demonstrate how this extension can be used to study the asymptotic average of the quadratic character of the resultant of polynomials over finite fields, answering a question of Ellenberg-Shusterman.

The Alexander polynomial can be constructed as an R-matrix invariant associated with representations of unrolled restricted quantum sl2 at a fourth root of unity. These highest representations V(t) are parameterized by nonzero complex numbers t which determine the polynomial variable. In this talk, we discuss a generalization to a multivariable link invariant computed from higher rank quantum groups at a fourth root of unity, with an emphasis on g=sl3.

We will review the representation theory of the sl2 case before moving to sl3. We then sketch a proof of the following theorem: For any knot K, the two-variable sl3 polynomial is equal to the Alexander polynomial when evaluated at (t1,t2) such that the representation V(t1,t2) is reducible. We compare the sl3 invariant with other knot invariants, such as the Jones polynomial, by giving specific examples. Unlike many well known knot invariants, the sl3 invariant can detect knot mutation.