Linear relations between writhe and minimal crossing number in Conway familiesof ideal knots and links

Abstract

We showed earlier how to predict the writhe of any rational knot or link in itsideal geometric configuration, or equivalently the average of the 3D writhe overstatistical ensembles of random configurations of a given knot or link (Cerf andStasiak 2000 Proc. Natl Acad. Sci. USA 97 3795). There is no general relationbetween the minimal crossing number of a knot and the writhe of its idealgeometric configuration. However, within individual families of knots linearrelations between minimal crossing number and writhe were observed (Katritch etal 1996 Nature 384 142). Here we present a method that allows us to express thewrithe as a linear function of the minimal crossing number within Conway familiesof knots and links in their ideal configuration. The slope of the lines andthe shift between any two lines with the same slope can be computed.