Graph-Links

TL;DR

This paper reviews Graph-Link Theory, translating knot theory concepts into graph language, constructing invariants, and demonstrating differences from virtual links, advancing the understanding of graph-based knot invariants.

Contribution

It introduces methods to translate classical and virtual knot theory into graph theory, constructing invariants and proving minimality theorems in this new setting.

Findings

Some graph-links are not equivalent to virtual links.

Constructed invariants for graph-knots and graph-links.

Proved minimality theorems and functorial mappings.

Abstract

The present paper is a review of the current state of Graph-Link Theory (graph-links are also closely related to homotopy classes of looped interlacement graphs), dealing with a generalisation of knots obtained by translating the Reidemeister moves for links into the language of intersection graphs of chord diagrams. In this paper we show how some methods of classical and virtual knot theory can be translated into the language of abstract graphs, and some theorems can be reproved and generalised to this graphical setting. We construct various invariants, prove certain minimality theorems and construct functorial mappings for graph-knots and graph-links. In this paper, we first show non-equivalence of some graph-links to virtual links.