quantum link invariants

A generalization of the Alexander polynomial from higher rank quantum groups

| chrisman.76

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.

References:

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

Rotational Virtual Links, Parity Polynomials and Quantum Link Invariants

| chrisman.76

Lou Kauffman (University of Illinois-Chicago & Novosibirsk State University)

This talk discusses virtual knot theory and rotational virtual knot theory. In virtual knot theory we introduce a virtual crossing in the diagrams along with over crossings and under crossings. Virtual crossings are artifacts of representing knots in higher genus surfaces as diagrams in the plane.

Virtual diagrammatic equivalence is the same as studying knots in thickened surfaces up to ambient isotopy, surface homeomorphisms and handle stabilization. At the diagrammatic level, virtual knot equivalence is generated by Reidemeister moves plus detour moves. In rotational virtual knot theory, the detour moves are restricted to regular homotopy of plane curves (the self-crossings are virtual). Rotational virtual knot theory has the property that all classical quantum link invariants extend to quantum invariants of rotational virtual knots and links. We explain this extension, and we consider the question of the power of quantum invariants in this context.

By considering first the bracket polynomial and its extension to a parity bracket polynomial for virtual knots (Manturov) and its further extension to a rotational parity bracket polynomial for knots and links (Kaestner and Kauffman), we give examples of links that are detected via the parity invariants that are not detectable by quantum invariants. In the course of the discussion we explain a functor from the rotational tangle category to the diagrammatic category of a quantum algebra. We delineate significant weaknesses in quantum invariants and how these gaps can be fulfilled by taking parity into account.

The slides for the talk are given below:

RotationalVirtualKnotsKeynote.key

References

Kauffman, Louis H, Rotational virtual knots and quantum link invariants, https://arxiv.org/abs/1509.00578