Introduction to Graph-Link Theory

TL;DR

This paper introduces graph-link theory as a combinatorial generalization of virtual knot theory, defining invariants like the Jones polynomial and extending classical theorems to this new framework.

Contribution

It develops the concept of graph-links, generalizes virtual links, and defines a Jones polynomial invariant within this new combinatorial setting.

Findings

Graph-links generalize virtual links.

Jones polynomial is defined and shown to be invariant for graph-links.

A generalization of the minimal diagram theorem is proved for graph-links.

Abstract

The present paper is an introduction to a combinatorial theory arising as a natural generalisation of classical and virtual knot theory. There is a way to encode links by a class of `realisable' graphs. When passing to generic graphs with the same equivalence relations we get `graph-links'. On one hand graph-links generalise the notion of virtual link, on the other hand they do not feel link mutations. We define the Jones polynomial for graph-links and prove its invariance. We also prove some a generalisation of the Kauffman-Murasugi-Thistlethwaite theorem on `minmal diagrams' for graph-links