On the highest degree of absolute polynomials of alternating links
type: Item
instance of: scholarly article
title: On the highest degree of absolute polynomials of alternating links
zbMATH Open document ID: 0639.57005
author: Tetsuo Miyauchi
published in: Proceedings of the Japan Academy. Series A
publication date: 1987-01-01T00:00:00Z
review text: Let \(c(\tilde L)\) denote the number of crossings of a regular projection \(\tilde L\) of a tame link L in \(S^ 3\), and \(c(L)\) denote the minimum number of crossings among all regular projections of L. Let \(Q(L)\) be the absolute polynomial which is an unoriented link invariant defined in [\textit{R. D. Brandt}, \textit{W. B. R. Lickorish} and \textit{K. C. Millett}: Invent. Math. 84, 563-573 (1986; Zbl 0595.57009)]. In this paper, the author shows the following as a corollary of his theorem: Corollary 1. If L is a prime alternating link then the highest degree of \(Q(L)\) is equal to \(c(L)-1\). Moreove he shows the following as a corollary of the same theorem which was independently shown by \textit{M. E. Kidwell} [On the degree of the Brandt-Lickorish-Millett polynomial of a link (preprint).] Corollary 2. If \(L_ 1\) and \(L_ 2\) are alternating links and \(L=L_ 1\#L_ 2\) is also an alternating link, then for any connected regular proper alternating projection \(\tilde L\) of L, \(g(\tilde L)\) and \(g^*(\tilde L)\) have cut-vertices. Where \(g(\tilde L)\) and \(g^*(\tilde L)\) are the graphs of \(\tilde L\) such that the vertices of \(g(\tilde L)\) and \(g^*(\tilde L)\) correspond to the white and the black domains, respectively, and each edge of \(g(\tilde L)\) and \(g^*(\tilde L)\) corresponds to a crossing of \(\tilde L\).
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 4042854
zbMATH Keywords: number of crossings of a regular projection of a tame link
zbMATH Keywords: absolute polynomial
zbMATH Keywords: prime alternating link
zbMATH Keywords: Brandt-Lickorish-Millett polynomial of a link
zbMATH Keywords: cut-vertices
reviewed by: Yasuhiro Nakagawa
MaRDI profile type: Publication
full work available at URL: pjaa.63.174
OpenAlex ID: W1989711849
cites work: A polynomial invariant for unoriented knots and links
cites work: A polynomial invariant for knots via von Neumann algebras
cites work: On the Degree of the Brandt-Lickorish-Millett-Ho Polynomial of a Link
cites work: Closed incompressible surfaces in alternating knot and link complements
cites work: Jones polynomials and classical conjectures in knot theory
DOI: 10.3792/PJAA.63.174
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description: scientific article; zbMATH DE number 4042854