Why the 3D Writhe of Ideal Knots and Links Is Quantized and Additive

Abstract

Besides mathematical interest, knots and knot theory have important applications in physics, chemistry, and biology. Stasiak and colleagues devised a constructive method for a knot "energy" using a Metropolis Monte Carlo algorithm to minimize the length of rope needed to realize a given knot for a rope of a fixed diameter. They called the resulting rendering of the knot the "ideal" configuration. They have found (i) that the 3-dimensional (3D) writhe of the ideal configuration of a knot appeared, empirically, to be quantized; (ii) empirically, the 3D writhe of the ideal configuration was additive over composition of knots; and (iii) the 3D writhe of ideal configuration of alternating knots (which have a projection in which crossings alternate between over and under as one transverses the knot in a fixed direction) seems to be equal, to 1*T + 3/7*(W-B), where T is the "Tait number," and W and B are the number of the white and black regions in the checkerboard coloring of the knot. It is still not clear why properties i-iii might be true. Here I show that all these empirical findings can be related to the fact that T and W-B can be proven to be additive over composition of alternating knots. In particular, (1) the 3D writhe of ideal configurations is likely only to be additive for composition of alternating knots, and (2) remarkably, the geometrical structure of all alternating (even prime) knots is determined by the three smallest knots.