A generalization of the Alexander polynomial from higher rank quantum groups

Matthew Harper (The Ohio State University)

Murakami and Ohtsuki have shown that the Alexander polynomial is an $R$ -matrix invariant associated with representations $V(t)$ of unrolled restricted quantum $\mathfrak{sl}_2$ at a fourth root of unity. In this context, the highest weight $t\in\mathbb{C}^\times$ of the representation determines the polynomial variable. In this talk, we discuss an extension of their construction to a link invariant $\Delta_{\mathfrak{g}}$ , which takes values in $n$ -variable Laurent polynomials, where $n$ is the rank of $\mathfrak{g}$ .

We begin with an overview of computing quantum invariants and of the $\mathfrak{sl}_2$ case. Our focus will then shift to $\mathfrak{g}=\mathfrak{sl}_3$ . After going through the construction, we briefly sketch the proof of the following theorem: For any knot $K$ , evaluating $\Delta_{\mathfrak{sl}_3}$ at ${t_1=\pm1}$ , ${t_2=\pm1}$ , or ${t_2=\pm it_1^{-1}}$ recovers the Alexander polynomial of $K$ . We also compare $\Delta_{\mathfrak{sl}_3}$ with other invariants by giving specific examples. In particular, this invariant can detect mutation and is non-trivial on Whitehead doubles.