Hard Diagrams of the Unknot

AbstractWe present three “hard” diagrams of the unknot. They require (at least) three extra crossings before they can be simplified to the trivial unknot diagram via Reidemeister moves in S2. Two of them are constructed by applying previously proposed methods. The proof of their hardness uses significant computational resources. We also determine that no small “standard” example of a hard unknot diagram requires more than one extra crossing for Reidemeister moves in S2.