Reachability under arc crossing changes

Abstract

Cericola proved that every knot diagram can be transformed into an ascending unknot diagram by arc crossing changes. We refine this result for all diagrams on a fixed R1-reduced classical knot shadow with more than three crossings. Apart from at most two source diagrams, any two diagrams of the same state parity are mutually reachable. Each source diagram reaches every non-source diagram of its parity but cannot be reached from another diagram. A local five-occurrence condition on the marked Gauss word characterizes the sources and thereby determines the full directed reachability relation.