Multi-Virtual Knot Theory

TL;DR

This paper introduces multi-virtual knot theory, a generalization involving multiple types of virtual crossings, motivated by graph theory and extending classical knot invariants to a broader setting.

Contribution

It develops the theory of multi-virtual knots and links, connecting graph-theoretic concepts with topological knot theory, and generalizes Penrose evaluations for graph colorings.

Findings

Defines multi-virtual knot and link structures

Extends Penrose evaluation to new graph classes

Links graph theory with topological knot invariants

Abstract

This paper discusses a generalization of virtual knot theory that we call multi-virtual knot theory. Multi-virtual knot theory uses a multiplicity of types of virtual crossings. As we will explain, this multiplicity is motivated by the way it arises first in a graph-theoretic setting in relation to generalizing the Penrose evaluation for colorings of planar trivalent graphs to all trivalent graphs, and later by its uses in a virtual knot theory. As a consequence, the paper begins with the graph theory as a basis for our constructions, and then proceeds to the topology of multi-virtual knots and links. The second section of the paper is a review of our previous work (See arXiv:1511.06844). The reader interested in seeing our generalizations of the original Penrose evaluation, can begin this paper at the beginning and see the graph theory. A reader primarily interested in multi-virtual knots and links can begin reading in section 4 with references to the earlier part of the paper.