Immersed disks, slicing numbers and concordance unknotting numbers

Abstract

We study three knot invariants related to smoothly immersed disks in the four-ball. These are the four-ball crossing number, which is the minimal number of normal double points of such a disk bounded by a given knot; the slicing number, which is the minimal number of crossing changes to a slice knot; and the concordance unknotting number, which is the minimal unknotting number in a smooth concordance class. Using Heegaard Floer homology we obtain bounds that can be used to determine two of these invariants for all prime knots with crossing number ten or less, and to determine the concordance unknotting number for all but thirteen of these knots. As a further application we obtain some new bounds on Gordian distance between torus knots. We also give a strengthened version of Ozsvath and Szabo's obstruction to unknotting number one.

24 pages, 5 figures. V2: added section on Gordian distances between torus knots. V3: Improved exposition incorporating referees' suggestions. Accepted for publication in Comm. Anal. Geom

References in corpus (2)

• Semigroups, d-invariants and deformations of cuspidal singular points of plane curves
• Concordance properties of parallel links

Cited by in corpus (8)

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• Topologically slice knots that are not smoothly slice in any definite 4-manifold
• On the First Singularity for the Upsilon Invariant of Algebraic Knots
• Untwisting information from Heegaard Floer homology
• Knots that are not slice either in positons or in negatons