Local moves for links with common sublinks

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.