Rotational Virtual Knots and Quantum Link Invariants

TL;DR

This paper explores rotational virtual knot theory, extending quantum link invariants to this setting, and demonstrates how certain non-trivial links are undetectable by these invariants, highlighting both the potential and limitations of quantum algebra methods.

Contribution

It introduces an extension of quantum invariants to rotational virtual knots and links, and analyzes their effectiveness and limitations through algebraic and diagrammatic methods.

Findings

Extended the bracket polynomial and parity bracket for rotational virtual knots.

Identified non-trivial rotational links undetectable by quantum invariants.

Showed the naturality of quantum invariants within Hopf algebra frameworks.

Abstract

This paper studies 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. The paper sets up the background virtual knot theory, defines rotational virtual knot theory, studies an extension of the bracket polynomial and the Manturov parity bracket for rotationals. We give examples of links that are not detected by the bracket polynomial but are detected by the extended parity bracket. Then the general frameworks for oriented and unoriented quantum invariants are introduced and formulated for rotational virtual links. The paper ends with a section on 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 non-trivial link L has trivial functorial image. This means that this link is not detected by any quantum invariant formulated as outlined in this paper. We show earlier in the paper that the link L is a non-trivial rotational link using the parity bracket. 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. There are inherent limitations to studying rotational virtual knots by quantum algebra alone.