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

Why unoriented Khovanov homology is useful in graph theory

Scott Baldridge (LSU)

The Jones polynomial and Khovanov homology of a classical link are oriented link invariants—they depend upon an initial choice of orientation for the link. In this talk, we describe a Jones polynomial and Khovanov homology theory for unoriented virtual links. We then show how to transfer these unoriented versions over to graph theory to construct the Tait polynomial of a trivalent graph. This invariant polynomial counts the number of 3-edge colorings of a graph when evaluated at 1. If this count is nonzero for all bridgeless planar trivalent graphs, then the famous four color theorem is true. Thus, we show how topological ideas can be used to have an impact in graph theory. This is joint work with Lou Kauffman and Ben McCarty.


Baldridge, S., “A cohomology theory for planar trivalent graphs with perfect matchings”, https://arxiv.org/abs/1810.07302

Baldridge, S., Lowrance, A., McCarty, B., “The 2-factor polynomial detects even perfect matchings”, https://arxiv.org/abs/1812.10346

Baldridge, S., Kauffman, L. H., McCarty, B., “Unoriented Khovanov homology”, https://arxiv.org/abs/2001.04512


Knot groups and virtual knots

H. A. Dye (Mckendree U) and A. Kaestner (North Park U)

In the paper, Virtual parity Alexander polynomials, we defined a virtual knot group that used information about the parity of the classical crossings. This virtual knot group was defined using ad-hoc methods. In the paper, Virtual knot groups and almost classical knots, Boden et al describe several different knot groups obtained from virtual knots. These knot groups are related and specializations lead to the classical knot group. Here, we construct a formal structure for virtual knot groups and examine specializations and extensions of the groups.


H. A. Dye and A. Kaestner, Virtual parity Alexander polynomials, https://arxiv.org/abs/1907.08709