A simple proof of the Crowell–Murasugi theorem

Kindred, Thomas

doi:10.2140/agt.2024.24.2779

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A simple proof of the Crowell–Murasugi theorem

Thomas Kindred

Algebraic & Geometric Topology 24 (2024) 2779–2785

DOI: 10.2140/agt.2024.24.2779

Abstract

We give an elementary, self-contained proof of the theorem, proven independently in 1958–1959 by Crowell and Murasugi, that the genus of any oriented nonsplit alternating link equals half the breadth of its Alexander polynomial (with a correction term for the number of link components), and that applying Seifert’s algorithm to any oriented connected alternating link diagram gives a surface of minimal genus.

Keywords

Seifert surface, Alexander polynomial, alternating link, alternating knot, Murasugi sum, plumbing, fiber surface, de-plumbing, knot genus, link genus, 3–genus, Seifert's algorithm, homogeneous link

Mathematical Subject Classification

Primary: 57K10, 57K14

References

Publication

Received: 10 October 2022

Revised: 9 February 2023

Accepted: 7 May 2023

Published: 19 August 2024

Authors

Thomas Kindred
Department of Mathematics Wake Forest University Winston-Salem, NC United States
http://www.thomaskindred.com

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Volume 24, issue 5 (2024)