A Celtic Framework for Knots and Links

Abstract

We describe a variant of a method used by modern graphic artists to design what are traditionally called Celtic knots, which are part of a larger family of designs called “mirror curves.” It is easily proved that every such design specifies an alternating projection of a link. We use medial graphs and graph minors to prove, conversely, that every alternating projection of a link is topologically equivalent to some Celtic link, specifiable by this method. We view Celtic representations of knots as a framework for organizing the study of knots, rather like knot mosaics or braid representations. The formalism of Celtic design suggests some new geometric invariants of links and some new recursively specifiable sequences of links. It also leads us to explore new variations of problems regarding such sequences, including calculating formulae for infinite sequences of knot polynomials. This involves a confluence of ideas from knot theory, topological graph theory, and the theory of orthogonal graph drawings.