The Combinatorial Revolution in Knot Theory

K not theory is usually understood to be the study of embeddings of topological spaces in other topological spaces. Classical knot theory, in particular, is concerned with the ways in which a circle or a disjoint union of circles can be embedded in R. Knots are usually described via knot diagrams, projections of the knot onto a plane with breaks at crossing points to indicate which strand passes over and which passes under, as in Figure 1. However, much as the concept of “numbers” has evolved over time from its original meaning of cardinalities of finite sets to include ratios, equivalence classes of rational Cauchy sequences, roots of polynomials, and more, the classical concept of “knots” has recently undergone its own evolutionary generalization. Instead of thinking of knots topologically as ambient isotopy classes of embedded circles or geometrically as simple closed curves in R, a new approach defines knots combinatorially as equivalence classes of knot diagrams under an equivalence relation determined by certain diagrammatic moves. No longer merely symbols standing in for topological or geometric objects, the knot diagrams themselves have become mathematical objects of interest. Doing knot theory in terms of knot diagrams, of course, is nothing new; the Reidemeister moves date back to the 1920s [21], and identifying knot invariants (functions used to distinguish different knot types) by checking invariance under the moves has been common ever since. A recent shift toward taking the combinatorial approach more