Thistlethwaite Theorems for Knotoids and Linkoids

TL;DR

This paper extends the classical Thistlethwaite theorem to knotoids, relating their invariants to ribbon graph polynomials, and introduces new concepts like twisted knotoids and marked ribbon graphs for these generalizations.

Contribution

It generalizes the Thistlethwaite theorem to knotoids on various surfaces, connecting their invariants to Bollobás-Riordan polynomials and defining new related structures.

Findings

Extended Thistlethwaite theorem to twisted arrow polynomial of knotoids.

Defined twisted knotoids and marked ribbon graphs.

Connected knotoid invariants to Bollobás-Riordan polynomial evaluations.

Abstract

The classical Thistlethwaite theorem for links can be phrased as asserting that the Kauffman bracket of a link can be obtained from an evaluation of the Bollob\'as-Riordan polynomial of a ribbon graph associated to one of the link's Kauffman states. In this paper, we extend this result to knotoids, which are a generalization of knots that naturally arises in the study of protein topology. Specifically we extend the Thistlethwaite theorem to the twisted arrow polynomial of knotoids, which is an invariant of knotoids on compact, not necessarily orientable, surfaces. To this end, we define twisted knotoids, marked ribbon graphs, and their arrow- and Bollob\'as-Riordan polynomials. We also extend the Thistlethwaite theorem to the loop arrow polynomial of knotoids in the plane, and to spherical linkoids.