Tied Links

TL;DR

This paper introduces tied links, a new mathematical object combining links with ties, and develops invariants and theorems extending classical braid and link theory to this new setting.

Contribution

It defines tied links and tied braids, introduces an invariant polynomial via skein relations, and proves Alexander and Markov theorems for tied links.

Findings

Defined tied links and tied braids.

Established an invariant polynomial for tied links.

Proved Alexander and Markov theorems for 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 remaining generators can be interpreted as ties between consecutive strands; this interpretation leads 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 that we have defined can be obtained also by using the Jones recipe.