Linear Behavior of the Writhe Versus the Number of Crossings in Rational Knots and Links

Abstract

Using the formula introduced in [Proc. Natl Acad. Sci. USA 97, 3795 (2000)], we can predict the 3D writhe of any rational knot or link in its ideal config- uration, or equivalently, the ensemble average of the 3D writhe of random configura- tions of it. Here we present a method that allows us to express the writhe as a linear function of the minimal crossing number within individual Conway families of ratio- nal knots and links. We discuss the cases of families with slopes ±4/7,±10/7,±1, and 0. For families with the same slope value, the vertical shift between the corre- sponding lines can also be computed.