A complete list of minimal diagrams of an oriented alternating knot

Abstract

By the Tait Flyping Conjecture, all possible minimal diagrams of an alternating knot are related by a finite sequence of flypes. We use this fact for an algorithm to produce all possible minimal diagrams of an oriented alternating prime knot type. Starting with any given oriented minimal alternating diagram of a prime knot, the algorithm performs flypes exhaustively until all other possible minimal diagrams of the oriented knot type have been produced. Results for all distinct alternating prime knot types up to 16 crossings are shown. In addition, prime knot families are discussed that have either a small or a very large number of distinct minimal diagrams.