Title: The Linearity of the Ropelenghts of Conway Algebraic Knots in Terms of Their Crossing Numbers
Abstract: For a knot or link K, let L(K) denote the ropelength of K and let Cr(K) denote the crossing number of K. An important problem in geometric knot theory concerns the relationship between L(K) and Cr(K) (or intuitively, the relationship between the length of a rope needed to tie a particular knot and the complexity of the knot). We show that there exists a constant a > 0 such that if a knot K allows a special knot diagram D (called Conway algebraic knot diagram) with n crossings, then L(K) a·n. Furthermore, if D is alternating (but not necessarily reduced and in fact K may not have a minimal alternating diagram that is algebraic), then L(K) a·Cr(K). The approach used here can be applied to a larger class of knots, namely those formed by replacing single crossings in a Conway algebraic knot diagram by tangles whose crossing number is bounded by a constant. Interestingly, it has been shown by the same authors that the Jones polynomials of these knots can be computed in polynomial time.
Publication Year: 2008
Publication Date: 2008-01-01
Language: en
Type: article
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