Minimal Complexity of C-Complexes

Abstract

In knot theory, a link is a disjoint union of circles (i.e. components) in 3-dimensional space. A goal of knot theory is to measure the interaction between the various components of a link. One measure of the complexity of a link is the complexity of a 2-dimensional object bounded by this link. One such object is a Ccomplex (or clasp-complex). We ask the question, “Given a link, what is the least number of clasps amongst all C-complexes bounded by that link?” For two-component links, we have found a precise formula for the minimal number of clasps. In the case of links of three components, we prove a bound in terms of a generalization of the classical linking number called the triple linking number, by relating this problem to minimal perimeter polyominoes.