Parity and Exotic Combinatorial Formulae for Finite-Type Invariants of Virtual Knots

TL;DR

This paper introduces new combinatorial formulae for virtual knot invariants using parity, revealing exotic properties and non-GPV finite-type invariants, with explicit examples and decompositions.

Contribution

It develops parity-based combinatorial formulae for virtual knots that exhibit exotic properties and are not of classical finite-type, expanding the understanding of virtual knot invariants.

Findings

Existence of integer-valued virtualization invariants of all orders.

Decomposition of Polyak-Viro formulae into even and odd parts.

Presentation of eleven new non-trivial order 2 invariants.

Abstract

The present paper produces examples of Gauss diagram formulae for virtual knot invariants which have no analogue in the classical knot case. These combinatorial formulae contain additional information about how a subdiagram is embedded in a virtual knot diagram. The additional information comes from the second author's recently discovered notion of parity. For a parity of flat virtual knots, the new combinatorial formulae are Kauffman finite-type invariants. However, many of the combinatorial formulae possess exotic properties. It is shown that there exists an integer valued virtualization invariant combinatorial formula of order n for every n (i.e. it is stable under the map which changes the direction of one arrow but preserves the sign). Hence, it is not of Goussarov-Polyak-Viro finite-type. Moreover, every homogeneous Polyak-Viro combinatorial formula admits a decomposition into an "even" part and an "odd" part. For the Gaussian parity, neither part of the formula is of GPV finite-type when it is nonconstant on the set of classical knots. In addition, eleven new non-trivial combinatorial formulae of order 2 are presented which are not of GPV finite-type.