Electric group for knots and links

TL;DR

This paper introduces a new approach to electric groups for oriented knots and links, linking diagram colorings to group homomorphisms and proposing tensor network invariants akin to quantum invariants.

Contribution

It generalizes the electric invariant to oriented knots and links and introduces tensor network invariants, connecting diagram colorings with algebraic structures.

Findings

Homomorphisms correspond to proper diagram colorings.

Introduces tensor network invariants for colored links.

Provides a unified approach to electric groups for knots and links.

Abstract

In 2014 Andrey Perfiliev introduced the so-called electric invariant for non-oriented knots. This invariant was motivated by using Kirchhoff's laws for the dual graph of the knot diagram. Later, in 2020, Anastasiya Galkina generalised this invariant and defined the electric group for non-oriented knots. Both works were never written and published. In the present paper we describe a simple and general approach to the electric group for oriented knots and links. Each homomorphism from the electric group to an arbitrary finite group can be described by a proper colouring of the diagram. This colouring assigns an element of the group to each crossing of the diagram, and the proper conditions correspond to the areas of the diagram. In the second part of the paper we introduce tensor network invariants for coloured links. The idea of these invariants is very close to quantum invariants for classical links.