Alternating links and definite surfaces

TL;DR

This paper characterizes alternating links using definite surfaces, offers a new proof of Tait's conjecture on crossing number and writhe, and presents an exponential algorithm for recognizing alternating knots.

Contribution

It introduces a new characterization of alternating links via definite surfaces and applies it to prove classical conjectures and develop an algorithm.

Findings

Confirmed that reduced alternating diagrams have consistent crossing number and writhe.

Provided a new proof of Tait's conjecture using definite surfaces.

Developed an exponential time algorithm for recognizing alternating knots.

Abstract

We establish a characterization of alternating links in terms of definite spanning surfaces. We apply it to obtain a new proof of Tait's conjecture that reduced alternating diagrams of the same link have the same crossing number and writhe. We also deduce a result of Banks and Hirasawa-Sakuma about Seifert surfaces for special alternating links. The appendix, written by Juh\'asz and Lackenby, applies the characterization to derive an exponential time algorithm for alternating knot recognition.