The parity hierarchy and new invariants of knots in thickened surfaces

TL;DR

This paper introduces a new Laurent polynomial invariant for virtual knots in thickened surfaces, utilizing a modified Wirtinger presentation and the parity hierarchy to distinguish crossing types and enhance the invariant.

Contribution

It develops a novel invariant for virtual knots in thickened surfaces using parity concepts, extending previous invariants with a hierarchy-based enhancement.

Findings

Invariant is a Laurent polynomial in 2g+3 variables.

Invariant distinguishes between different crossing types.

Construction applies to virtual knots as well.

Abstract

In the present paper, we construct an invariant for virtual knots in the thickened sphere with g handles; this invariant is a Laurent polynomial in 2g+3 variables. To this end, we use a modification of the Wirtinger presentation of the knot group and the concept of parity introduced by V.O.Manturov. The section 4 of the paper is devoted to an enhancement of the invariant (construction of the invariant module) by using the parity hierarchy concept suggested by V.O.Manturov. Namely, we discriminate between odd crossings and two types of even crossings; the latter two types depend on whether an even crossing remains even/odd after all odd crossings of the diagram are removed. The construction of the invariant also works for virtual knots.