Rack shadows and their invariants

TL;DR

This paper introduces the concept of rack shadows, sets with rack actions, and demonstrates how shadow colorings of link diagrams can produce enhanced invariants that outperform traditional counting invariants.

Contribution

It defines rack shadows and develops enhanced invariants for classical links using shadow colorings, showing these are more powerful than unenhanced invariants.

Findings

Enhanced invariants are stronger than unenhanced counting invariants.

Shadow colorings provide a new method for link invariant enhancement.

The approach parallels vector space concepts over fields.

Abstract

A rack shadow is a set X with a rack action by a rack R, analogous to a vector space over a field. We use shadow colorings of classical link diagrams to define enhanced rack counting invariants and show that the enhanced invariants are stronger than unenhanced counting invariants.