Immersed disks, slicing numbers and concordance unknotting numbers

Owens, B. and Strle, S. (2016) Immersed disks, slicing numbers and concordance unknotting numbers. Communications in Analysis and Geometry, 24(5), pp. 1107-1138. (doi:10.4310/CAG.2016.v24.n5.a8)

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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.

Item Type:Articles
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Owens, Dr Brendan
Authors: Owens, B., and Strle, S.
Subjects:Q Science > QA Mathematics
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Communications in Analysis and Geometry
Journal Abbr.:Comm. Anal. Geom.
Publisher:International Press
ISSN:1019-8385
ISSN (Online):1944-9992
Copyright Holders:Copyright © 2016 International Press
First Published:First published in Communications in Analysis and Geometry 24(5):1107-1138
Publisher Policy:Reproduced in accordance with the copyright policy of the publisher
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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
564181Alternating links and cobordism groupsBrendan OwensEngineering & Physical Sciences Research Council (EPSRC)EP/I033754/1M&S - MATHEMATICS