Local moves for links with common sublinks

TL;DR

This paper investigates how local moves called C_k^d-moves relate links with common sublinks, showing they can connect certain complex links to trivial links and exploring implications for finite type invariants.

Contribution

It demonstrates that (n,k)-Brunnian links can be transformed into trivial links using C_k^d-moves, extending previous results and analyzing invariants.

Findings

(n,k)-Brunnian links can be deformed into trivial links via C_k^d-moves.

Provides counter-examples for general cases.

Results on finite type invariants of (n,k)-Brunnian links.

Abstract

A C_k-move is a local move that involves (k+1) strands of a link. A C_k-move is called a C_k^d-move if these (k+1) strands belong to mutually distinct components of a link. Since a C_k^d-move preserves all k-component sublinks of a link, we consider the converse implication: are two links with common k-component sublinks related by a sequence of C_k^d-moves? We show that the answer is yes under certain assumptions, and provide explicit counter-examples for more general situations. In particular, we consider (n,k)-Brunnian links, i.e. n-component links whose k-component sublinks are all trivial. We show that such links can be deformed into a trivial link by C_k^d-moves, thus generalizing a result of Habiro and Miyazawa-Yasuhara, and deduce some results on finite type invariants of (n,k)-Brunnian links.