|
Organizers |
Rotational Virtual Links and Quantum Link Invariants
by
Louis H Kauffman
University of Illinois at Chicago
This talk is about rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. We set up the background virtual knot theory, define rotational virtual knot theory, study an extension of the bracket polynomial and the Manturov parity bracket for rotationals. Then the general frameworks for oriented and unoriented quantum invariants are introduced and formulated for rotational virtual links. The talk ends with quantum link invariants in the Hopf algebra framework where one can see the naturality of using regular homotopy combined with virtual crossings (permutation operators), as they occur significantly in the category associated with a Hopf algebra. We show how this approach via categories and quantum algebras illuminates the structure of invariants that we have already described via state summations. In particular, we show that a certain link L has trivial functorial image. This means that this link is not detected by any quantum invariant formulated as outlined here. At this writing, we conjecture that the link L is a non-trivial rotational link. These calculations with the diagrammatic images in quantum algebra show how this category forms a higher level language for understanding rotational virtual knots and links. If the conjecture is true, then there are inherent limitations to studying rotational virtual knots by quantum algebra alone.
Date received: October 19, 2015
Copyright © 2015 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cblw-11.