Mathematics Colloquia and Seminars

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Twenty years ago Scharlemann and Thompson used deep results from sutured manifold theory to prove that a genus reducing crossing change on a knot maybe be realized as untwisting a Hopf band plumbed onto a minimal genus Seifert surface. This gives a hint that understanding genus reducing crossing changes is closely related to understanding how a compact surface in S^3 changes when it is twisted. In this talk we use modern technology from the theory of Heegaard splittings to show that understanding when two surfaces are related by a single twist implies the existence of an algorithm to determine when two (hyperbolic or fibered) knots of different genus are related by a single crossing change.