Tied links and invariants for singular links

TL;DR

This paper develops an algebraic-combinatoric framework for tied links, proves a decomposition of the tied braid monoid, and introduces new polynomial invariants for tied singular links that distinguish links beyond existing invariants.

Contribution

It provides a purely algebraic-combinatoric version of tied links, proves a decomposition theorem for the tied braid monoid, and introduces new invariants for tied singular links.

Findings

Decomposition of tied braid monoid as a semi-direct product.

Reproof of Alexander and Markov theorems for tied links.

New five-variable polynomial invariants that distinguish certain singular links.

Abstract

Tied links and the tied braid monoid were introduced recently by the authors and used to define new invariants for classical links. Here, we give a version purely algebraic-combinatoric of tied links. With this new version we prove that the tied braid monoid has a decomposition like a semi--direct group product. By using this decomposition we reprove the Alexander and Markov theorem for tied links; also, we introduce the tied singular knots, the tied singular braid monoid and certain families of Homflypt type invariants for tied singular links; these invariants are five-variables polynomials. Finally, we study the behavior of these invariants; in particular, we show that our invariants distinguish non isotopic singular links indistinguishable by the Paris-Rabenda invariant.