Starred Knots, Knotoids, Virtual Knots and Mock Alexander Polynomials



2022 12 05

15:30 (GMT+0)


Louis Kauffman

Knots and representation theory

This talk is joint work with Neslihan Gugumcu and continues the talk from last week.
A "starred knot" is a knot or link diagram with "stars" placed in some of its regions. Reidemeister moves are not allowed to pass arcs across the stars. The starred knot is taken up to this restricted form of Reidemeister equivalence. Adding the star is equivalent to removing a tube from either the thickened plane or from the thickened two-sphere (or from a thckened surface when we generalize to virtual knots). Thus we are using diagrammatic models for knots and links in handle bodies.
This talk will discuss generalizations of the Alexander-Conway polynomial to starred knots, knotoids and virtual knots and knots in thickened surfaces.
These Mock Alexander Polynomials can be defined on diagrams whenever the number of faces in the diagram is equal to the number of crossings.
The generalizations use state summations that can be expressed in terms of permanents of matrices associated with the diagrams of the starred entities. These state summations generalize the structures in the author's monograph Formal Knot Theory that apply to the Alexander-Conway polynomial. Thus we create analogs of the Alexander-Conway polynomial via state summation. We will show by examples how the state sums we are using go beyond the determinant formulations of the the original Alexander polynomial and we will give a number of new examples of computations and formulations of the invariants.

Time line

Chicago -8h

Moscow +0h

Beijing +5h

Seoul +6h

Tokyo +6h