Immersed disks, slicing numbers and concordance unknotting numbers

Abstract

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 required to obtain 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 Ozsváth and Szabó's obstruction to unknotting number one.

Unlinking information from 4-manifolds

“…The main result of this paper is a generalization of a theorem of Cochran and Lickorish [6], and of a refinement due to the second author and Strle [24], to the case of links with more than one component. We choose an orientation on a given link and consider the sum σ + η of the classical link signature and nullity.…”

“…The first author thanks the University of Glasgow for its hospitality. The second author acknowledges the influence of his earlier joint work with Sašo Strle, especially [24]. We thank Mark Powell for helpful comments on an earlier draft, and the anonymous referee for helpful suggestions.…”

Untwisting information from Heegaard Floer homology

“…In Section 4, we get around the failure of the Montesinos trick for untwisting number-1 knots by porting the machinery used by Owens and Strle in [OweS13] and Nagel and Owens in [NO15] as an obstruction to low untwisting number:…”

Signature invariants related to the unknotting number

“…Thus, they also bounds the number of crossing changes required to convert K into a slice knot (the slicing number of K) and the number of crossing changes required to convert K into an algebraically slice knot (the algebraic slicing number). Past work on these invariants includes [22,31,32].…”