Examples relating to the crossing number, writhe, and maximal bridge length of knot diagrams
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 crossing number \(c\), they prove that, if \(\text{deg}_zF(a,z)=c-2\), then \(\text{deg }Q(z)=c-2\); if \(\text{deg}_zF(a,z)=c-3\) and \(\text{deg }Q(z)<c-3\), then \(K\) is semiadequate (or adequate). Furthermore, the authors describe a construction in order to show that any link has a diagram of maximal bridge length 2. This diagram can be made to have no nugatory crossings or trivial clasps. Of course, this construction is made possible at the cost of high augmentation of the crossing number of the diagram. Finally, some open problems concerning the writhe variation of a knot \(K\) and the relation between the degree of the Kauffman polynomial of \(K\) and the degree of the HOMFLY polynomial of a Whitehead double of \(K\) complete the paper.
scientific article; zbMATH DE number 1943241
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