Virtual Knot Invariants Arising From Parities

TL;DR

This paper generalizes the concept of parity in knot theory, introduces parities with coefficients from abelian groups, and explores their universal properties and connections to homology, leading to new invariants and insights for different knot theories.

Contribution

It extends the notion of parity to abelian group coefficients, identifies universal parities, and links parities in flat knots to homology groups, advancing the understanding of knot invariants.

Findings

Existence of universal parity for certain knot theories

All parities in flat knots derive from homology groups

Gaussian parity is the only non-trivial parity for free knots

Abstract

In \cite {FrKn,Sbornik} it was shown that in some knot theories the crucial role is played by {\em parity}, i.e.\ a function on crossings valued in $\{0,1\}$ and behaving nicely with respect to Reidemeister moves. Any parity allows one to construct functorial mappings from knots to knots, to refine many invariants and to prove minimality theorems for knots. In the present paper, we generalise the notion of parity and construct parities with coefficients from an abelian group rather than $\mathbb{Z}_2$ and investigate them for different knot theories. For some knot theories we show that there is the universal parity, i.e.\ such a parity that any other parity factors through it. We realise that in the case of flat knots all parities originate from homology groups of underlying surfaces and, at the same time, allow one to "localise" the global homological information about the ambient space at crossings. We prove that there is only one non-trivial parity for free knots, the Gaussian parity. At the end of the paper we analyse the behaviour of some invariants constructed for some modifications of parities.