The Classification of fused links

TL;DR

This paper introduces a complete invariant for classifying fused links, establishing a one-to-one correspondence between their equivalence classes and elements of a specific algebraic structure, up to symmetric group conjugation.

Contribution

It constructs the first complete invariant for fused links and characterizes their classification via algebraic structures and conjugation actions.

Findings

Complete invariant for fused links constructed

Classification corresponds to elements of abelization modulo conjugation

Provides a new algebraic approach to link classification

Abstract

We construct the complete invariant for fused links. It is proved that the set of equivalence classes of $n$-component fused links is in one-to-one correspondence with the set of elements of the abelization $UVP_n/UVP_n^{\prime}$ up to conjugation by the elements from the symmetric group $S_n<UVB_n$.