Genera of the alternating links
type: Item
instance of: scholarly article
title: Genera of the alternating links
zbMATH Open document ID: 0631.57004
DOI: 10.1215/S0012-7094-86-05336-6
author: David Gabai
published in: Duke Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: One of the fundamental results in link theory is \textit{H. Seifert}'s [Math. Ann. 110, 571-592 (1934; Zbl 0010.13303)] constructive proof of the fact that every tame link in the three-sphere is the oriented boundary of an orientable surface. It is a theorem of \textit{R. H. Crowell} [Ann. Math., II. Ser. 69, 258-275 (1959; Zbl 0111.358)] and \textit{K. Murasugi} [J. Math. Soc. Japan 10, 94-105 (1958; Zbl 0084.193); ibid. 235-248 (1958; Zbl 0106.167)] that Seifert's construction, applied to an alternating link diagram, always results in a spanning surface that is of minimal genus for that link type. Their proofs depend on the relationship between the degree of the Alexander polynomial and the genus of a spanning surface. In this paper the author presents a very different proof of this theorem, using elementary homology theory and geometric arguments. Regrettably, the closing passages of the paper leave this reader uncertain about several important aspects of the proof.
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 4026349
zbMATH Keywords: genus of links
zbMATH Keywords: Seifert surface
zbMATH Keywords: alternating link
zbMATH Keywords: spanning surface
MaRDI profile type: Publication
cites work: Genus of alternating link types
cites work: Foliations and genera of links
cites work: Foliations and the topology of 3-manifolds
cites work: The Murasugi sum is a natural geometric operation
cites work: On the genus of the alternating knot. I
cites work: On the genus of the alternating knot. II
cites work: Q3709485
full work available at URL: s0012-7094-86-05336-6
OpenAlex ID: W1991352225
description: scientific article; zbMATH DE number 4026349