The 250 Knots with up to 10 Crossings

Abstract

The list of knots with up to 10 crossings is commonly referred to as the Rolfsen Table. This paper presents a way to generate the Rolfsen table in a simple, clear, and reproducible manner. The methods we use are similar to those used by J. Hoste, M. Thistlethwaite, and J. Weeks in [1]. The difference between our methods comes from the fact that [1] uses a more complicated algorithm to be able to find all the knots with up to 17 crossings, while our approach demonstrates a simpler way to find the knots up to 10 crossings. We do this by generating all planar knot diagrams with up to 10 crossings and applying several simplifications to group the knot diagrams into equivalence classes. From these classes, we generate the full list of candidate knots and reduce it with several sets of moves. Lastly, we use invariants to show that each of the 250 diagrams generated is distinct, proving that there are exactly 250 knots with 10 crossings or fewer. Though the algorithms used could be made more efficient, readability was chosen over speed for simplicity and reproducibility.