Combinatorial knot diagrams

23 May 2025

The question this article seeks to answer is, how can one define a knot diagram “combinatorially”, that is, entirely in terms of finite sets and functions (so without reference to topological notions like the plane or the sphere)? Such a definition would allow one to represent knot diagrams on a computer and compute various knot invariants. It would also allow for an entirely combinatorial approach to knot theory, by defining a “knot” as a knot diagram up to Reidemeister moves. This question has certainly been answered before

combinatorial map