Tied links

Abstract

In this paper, we introduce the tied links, i.e. ordinary links provided with some “ties” between strands. The motivation for introducing such objects originates from a diagrammatical interpretation of the defining generators of the so-called algebra of braids and ties; indeed, one half of such generators can be interpreted as the usual generators of the braid algebra, and the other half can be interpreted as ties between consecutive strands; this interpretation leads to the definition of tied braids. We define an invariant polynomial for the tied links via a skein relation. Furthermore, we introduce the monoid of tied braids and we prove the corresponding theorems of Alexander and Markov for tied links. Finally, we prove that the invariant of tied links we defined can be obtained also by using the Jones recipe.