Three approaches to a bracket polynomial for singular links. (English) Zbl 1355.57004

Caprau, Carmen; Chichester, Alex; Chu, Patrick

Involve 10, No. 2, 197-218 (2017).

The Kauffman bracket is a polynomial invariant for unoriented knots and links and is defined via a skein relation, i.e., an identity involving knot (or singular link) diagrams that are the same except in a small neighborhood. In the paper under review the authors extend the Kauffman bracket to singular links (Section \(2\)). After that, in Section \(3\) they study the behavior of the extended Kauffman bracket with respect to disjoint unions, connected sums, and mirror images of singular links. In Section \(4\) they define a representation of the singular braid monoid into the Temperley-Lieb algebra, and use it to provide a different approach to the extended Kauffman bracket for singular links. Finally, in Section \(5\) they show how to arrive at the extended Kauffman bracket by interpreting singular link diagrams as abstract tensor diagrams and employing a solution to the Yang-Baxter equation.

Reviewer: J. N. Alonso Alvarez (Vigo)

MSC:

57M25	Knots and links in the \(3\)-sphere (MSC2010)
57M27	Invariants of knots and \(3\)-manifolds (MSC2010)

Keywords:

Kauffman bracket; invariants for knots and links; singular braids and links; Yang-Baxter equation