Singular Links and Yang-Baxter State Models

TL;DR

This paper develops a Yang-Baxter state model to create invariants for classical and singular links, extending existing polynomials and exploring their algebraic and topological properties.

Contribution

It introduces a novel Yang-Baxter state model for singular links and knotted graphs, expanding the scope of link invariants and their algebraic representations.

Findings

Constructed invariants for singular links and knotted graphs.

Extended the sl(n) polynomial to singular and oriented knotted objects.

Connected the invariant to the Murakami-Ohtsuki-Yamada state model.

Abstract

We employ a solution of the Yang-Baxter equation to construct invariants for knot-like objects. Specifically, we consider a Yang-Baxter state model for the sl(n) polynomial of classical links and extend it to oriented singular links and balanced oriented 4-valent knotted graphs with rigid vertices. We also define a representation of the singular braid monoid into a matrix algebra, and seek conditions for extending further the invariant to contain topological knotted graphs. In addition, we show that the resulting Yang-Baxter-type invariant for singular links yields a version of the Murakami-Ohtsuki-Yamada state model for the sl(n) polynomial for classical links.