On the classification of links up to finite type

TL;DR

This paper introduces a new approach to classify links up to finite type invariants by using an action of string links, leading to a clearer understanding of link equivalence classes and their structure.

Contribution

It defines an action of string links to lift indeterminacy in finite type invariants and characterizes link classes via orbit spaces and stabilizers.

Findings

Links up to the new indeterminacy correspond to orbit spaces of the action.

Structure theorems for C_n-equivalence and Self-C_n-equivalence are established.

Provides a bijection between link classes and certain orbit spaces.

Abstract

We use an action, of 2l-component string links on l-component string links, defined by the first author and Xiao-Song Lin, to lift the indeterminacy of finite type link invariants. The set of links up to this new indeterminacy is in bijection with the orbit space of the restriction of this action to the stabilizer of the identity. Structure theorems for the sets of links up to C_n-equivalence and Self-C_n-equivalence are also given.