The classification of links up to clasp-pass moves

TL;DR

This paper provides a comprehensive classification of links under clasp-pass moves and related moves, using clasper theory, extending previous partial results in link equivalence classifications.

Contribution

It offers a complete classification of links up to clasp-pass moves and related moves, generalizing earlier partial results with new proofs based on clasper theory.

Findings

Links are classified up to clasp-pass moves coinciding with Habiro's C_3-equivalence.

Classification up to band-pass and band-# moves is achieved, generalizing previous results.

Proofs utilize clasper theory to establish the classifications.

Abstract

We give a complete classification of links up to clasp-pass moves, which coincides with Habiro's $C_3$-equivalence. We also classify links up to band-pass and band-# moves, which are versions of the usual pass- and #-move, respectively, where each pair of parallel strands belong to the same component. This recovers and generalizes widely a number of partial results in the study of these local moves. The proofs make use of clasper theory.