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.


Harper, Matthew, “Verma Modules over Restricted \mathfrak{sl}_3 at a fourth root of unity”, https://arxiv.org/abs/1911.00641

Harper, Matthew, “A generalization of the Alexander Polynomial from higher rank quantum groups”, https://arxiv.org/abs/2008.06983

One thought on “A generalization of the Alexander polynomial from higher rank quantum groups

  1. Thank you, Matthew, for a fascinating talk! If you would like to ask a question of our speaker, please reply to this comment below.

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