Examples relating to the crossing number, writhe, and maximal bridge length of knot diagrams
type: Item
instance of: scholarly article
title: Examples relating to the crossing number, writhe, and maximal bridge length of knot diagrams
zbMATH Open document ID: 1030.57009
author: Mark E. Kidwell
author: Alexander Stoimenow
published in: Michigan Mathematical Journal
publication date: 2003-06-30T00:00:00Z
review text: In the 1980s, \textit{R. D. Brandt}, \textit{W. B. R. Lickorish} and \textit{K. C. Millett} [Invent. Math. 84, 563-573 (1986; Zbl 0595.57009)] and \textit{C. F. Ho} [A new polynomial invariant for knots and links, Abstr. Am. Math. Soc. 6, 300 (1985)] defined a link polynomial, denoted by \(Q(z)\). \textit{M. E. Kidwell} [Proc. Am. Math. Soc. 100, 755-762 (1987; Zbl 0638.57004)] proved that, if a knot has a diagram with \(c\) crossings and its longest (over or under-) bridge has length \(d\), then \(\operatorname {deg}Q(z)\leq c-d\). The equality can be achieved for alternating knots and for all knots with minimal crossing number below 10. It has recently been demonstrated by the authors of this paper [see \textit{A. Stoimenow}, The crossing number and maximal bridge length of a knot diagram (with an appendix by M. Kidwell), Pacific J. Math., to appear] that the previous relation is a strict inequality for every diagram of the ``Perko pair'' knot \(10_{161}=10_{162}\) [according to the notation of \textit{D. Rolfsen}, Knots and links, Publish or Perish, Houston, TX (1990; Zbl 0854.57002)]. This knot got its name and fame when \textit{K. A. Perko} [Proc. Am. Math. Soc. 45, 262-266 (1974; Zbl 0287.55003)] discovered that it refutes a conjecture of Tait: it possesses two 10-crossing diagrams (the minimum possible) with distinct writhes. \textit{L. H. Kauffman} [ Trans. Am. Math. Soc. 318, 417-471 (1990; Zbl 0763.57004)] generalized \(Q(z)\) to his two-variable polynomial \(F(a,z)\), and Thistlethwaite investigated its degree properties. In particular, he proved an analogous inequality to the previous one: \(\text{deg}_zF(a,z)\leq c-d\). A search on the computer program KnotScape has shown that \(c-d\) is not necessarily minimized with minimal \(c\). In this paper, the authors find an upper bound for the degree of the Kauffman polynomial of inadequate and semiadequate links in terms of minimal crossing number and writhe variation. In the case of a knot \(K\) with minimal…
reviewed by: Fulvia Spaggiari
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 1943241
zbMATH Keywords: knot
zbMATH Keywords: link
zbMATH Keywords: crossing number
zbMATH Keywords: writhe
zbMATH Keywords: Brandt-Lickorish-Millett-Ho polynomial
zbMATH Keywords: Kauffman polynomial
describes a project that uses: Knotscape
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1049832889
OpenAlex ID: W1997795318
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DOI: 10.1307/MMJ/1049832889
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description: scientific article; zbMATH DE number 1943241