A simple proof of the Crowell-Murasugi theorem

TL;DR

This paper provides an elementary, self-contained proof of the Crowell-Murasugi theorem, establishing that for alternating knots, the genus equals half the Alexander polynomial's breadth and Seifert's algorithm yields a minimal genus surface.

Contribution

It offers a simplified, accessible proof of a classical result, making the theorem more understandable and easier to verify.

Findings

Proves the genus of alternating knots equals half the Alexander polynomial's breadth.

Shows Seifert's algorithm produces minimal genus surfaces for alternating knots.

Simplifies the proof of a well-known theorem in knot theory.

Abstract

We give an elementary, self-contained proof of the theorem, proven independently in 1958-9 by Crowell and Murasugi, that the genus of an alternating knot equals half the breadth of its Alexander polynomial, and that applying Seifert's algorithm to any alternating knot diagram gives a surface of minimal genus.