Twisted intersection colorings, invariants and double coverings of twisted links

TL;DR

This paper introduces twisted intersection colorings for twisted links, constructs new invariants, and explores the relationship between twisted links and their double coverings, revealing infinitely many non-equivalent links with equivalent double coverings.

Contribution

It presents novel twisted intersection colorings and invariants for twisted links, and demonstrates their application in distinguishing links via double coverings.

Findings

Existence of infinitely many non-equivalent twisted links with equivalent double coverings

Introduction of a new method to construct twisted links with equivalent double coverings

Development of twisted intersection colorings as invariants

Abstract

Twisted links are a generalization of classical links and correspond to stably equivalence classes of links in thickened surfaces. In this paper we introduce twisted intersection colorings of a diagram and construct two invariants of a twisted link using such colorings. As an application, we show that there exist infinitely many pairs of twisted links such that for each pair the two twisted links are not equivalent but their double coverings are equivalent. We also introduce a method of constructing a pair of twisted links whose double coverings are equivalent.