Oriented Toral Knots Lattices
Russell B. Walker · Proceedings of the American Mathematical Society · 1983
Abstract. The common intersections between two sets of toral knots, having differing toral knot types and differing sequences of orientations, are removed. The remaining arc-segments are reconnected in a prescribed manner. In closed form, the number of " resulting loops " is provided. A question concerning two foliations of the 2-torus, one containing (p, ¿7)-Reeb components, the other (r, s)-Reeb components, leads in a natural way to a problem involving toral knot splicing and recombination [2,3): One set of n pairwise nonintersecting type (p, q) toral knots (the knot wraps p times longitudinally while q times meridionally) is given the orientation o1 G n,"=, { + 1,-1}- Here +1 refers to one possible (p, ^-"direction",-1 the other. Likewise, a set of m type (r, i)-toral knots ((r, s) ¥=(p, q)) is given the orientation, o2 E n,"L| { +1,-1}- The two sets are assumed to have the minimum number of intersection points, nm \\qr — ps \\. These points are removed and the remaining intervals reconnected according to Figure 1. A resulting system of toral knots and null-homotopic loops are formed by this procedure. Figure 1 Problem. What is the number and type of these resulting toral knots and how many such loops exist? One imagines a "toral city " having two sets of one-way toral knot "streets". If resident "drivers " are required to turn at every corner, how many possible "routes" exist and what are their types. Of course, the sequence as well as the number of "left onlys " and "right onlys " will effect the question's answer. In this note, an answer in closed form is provided Oriented toral knot systems. The described collection of oriented (p,q)- and (r, i)-knots is an oriented toral knot lattice and is denoted by L. The associated Received by the editors October 5, 1981.