An extension of the classical Thistlethwaite theorem for links asserts that the Kauffman bracket of a link can be obtained from an evaluation of the Bollobás-Riordan polynomial of a ribbon graph associated to one of the link’s Kauffman states. In this paper, we further extend this result to knotoids, which are a generalization of knots that naturally arise in applications such as DNA and 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 Bollobás-Riordan polynomials. We also extend the Thistlethwaite theorem to the loop arrow polynomial of knotoids in the plane, and to spherical linkoids.
Deng et al. (Fri,) studied this question.