Non-additivity of the unknotting number

Hans Boden
McMaster

Abstract

Every knot diagram can be converted into an unknot diagram by applying crossing changes. For a given knot, the minimum number of crossing changes needed, taken over all representative diagrams, is the unknotting number of that knot. In 1937 Wendt studied the unknotting number of composite knots, namely those of the form K # J. He posited that the unknotting number of K # J should be equal to the sum of the unknotting numbers of K and and that of J. Early evidence in support of the conjecture came from Marty Scharlemann, who in 1985 proved it for knots K, J with unknotting number one. The goal of the talk is to survey two recent preprints disproving the conjecture. The preprints are due to Mark Brittenham and Susan Hermiller, and their discovery has been a major breakthrough and suggests that knot theorists really do not understand unknotting at all well!