Essential state surfaces for knots and links

TL;DR

This paper introduces a canonical spanning surface derived from knot or link diagrams based on Kauffman states, providing conditions for its essentiality and applications in determining knot triviality and splittability.

Contribution

It extends the concept of essential surfaces to a broad class of knots and links, including semiadequate, homogeneous, and algebraic types, and generalizes Gabai's Murasugi sum theorem to nonorientable surfaces.

Findings

Provides a sufficient condition for the surface to be essential.

Enables detection of triviality and splittability from diagrams.

Extends Murasugi sum theorem to nonorientable surfaces.

Abstract

We study a canonical spanning surface obtained from a knot or link diagram depending on a given Kauffman state, and give a sufficient condition for the surface to be essential. By using the essential surface, we can see the triviality and splittability of a knot or link from its diagrams. This has been done on the extended knot or link class which includes all of semiadequate, homogeneous, and most of algebraic knots and links. In the process of the proof of main theorem, Gabai's Murasugi sum theorem is extended to the case of nonorientable spanning surfaces.