Intersection formulas for parities on virtual knots

TL;DR

This paper establishes a new theoretical framework linking parities on virtual knots to invariant cycles and quasi-indices, leading to the creation of new parity invariants for virtual knots.

Contribution

It introduces a novel connection between parities, invariant cycles, and quasi-indices, enabling the definition of new parity invariants for virtual knots.

Findings

Parities on virtual knots originate from invariant 1-cycles on diagram arcs.

Invariant cycles are characterized by quasi-indices at crossings.

A new series of parities for virtual knots is constructed.

Abstract

We prove that parities on virtual knots come from invariant 1-cycles on the arcs of knot diagrams. In turn, the invariant cycles are determined by quasi-indices on the crossings of the diagrams. The found connection between the parities and the (quasi)-indices allows to define a new series of parities on virtual knots.