Rotational Virtual Knots and Quantum Link Invariants

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.