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The list of knots with up to 10 crossings is commonly referred to as theRolfsen Table. This paper presents a way to generate the Rolfsen table in asimple, clear, and reproducible manner. The methods we use are similar to thoseused by J. Hoste, M. Thistlethwaite, and J. Weeks in [1]. The differencebetween our methods comes from the fact that [1] uses a more complicatedalgorithm to be able to find all the knots with up to 17 crossings, while ourapproach demonstrates a simpler way to find the knots up to 10 crossings. We dothis by generating all planar knot diagrams with up to 10 crossings andapplying several simplifications to group the knot diagrams into equivalenceclasses. From these classes, we generate the full list of candidate knots andreduce it with several sets of moves. Lastly, we use invariants to show thateach of the 250 diagrams generated is distinct, proving that there are exactly250 knots with 10 crossings or fewer. Though the algorithms used could be mademore efficient, readability was chosen over speed for simplicity andreproducibility.

The 250 Knots with up to 10 Crossings