On closed 3-braids with unknotting number one

Abstract

We prove that if an alternating 3-braid knot has unknotting number one, then there must exist an unknotting crossing in any alternating diagram of it, and we enumerate such knots. The argument combines the obstruction to unknotting number one developed by Ozsváth and Szabó using Heegaard Floer homology, together with one coming from Donaldson's Theorem A.

29 pages, 4 figures

References in corpus (9)

• On the Floer homology of plumbed three-manifolds
• Floer homology and surface decompositions
• Corrigendum to "Knot Floer homology detects fibred knots"
• On the Khovanov and knot Floer homologies of quasi-alternating links
• Lens spaces, rational balls and the ribbon conjecture
• Knots with unknotting number 1 and essential Conway spheres
• Ozsvath-Szabo and Rasmussen invariants of cable knots
• The almost alternating diagrams of the trivial knot
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Cited by in corpus (4)

• The lens space realization problem
• The rational Witt class and the unknotting number of a knot
• Homologically thin, non-quasi-alternating links
• Turaev genus, knot signature, and the knot homology concordance invariants