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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On slicing invariants of knots


Author: Brendan Owens
Journal: Trans. Amer. Math. Soc. 362 (2010), 3095-3106
MSC (2000): Primary 57M25
Published electronically: August 13, 2009
MathSciNet review: 2592947
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Abstract: The slicing number of a knot, $ u_s(K)$, is the minimum number of crossing changes required to convert $ K$ to a slice knot. This invariant is bounded above by the unknotting number and below by the slice genus $ g_s(K)$. We show that for many knots, previous bounds on the unknotting number obtained by Ozsváth and Szabó and by the author in fact give bounds on the slicing number. Livingston defined another invariant $ U_s(K)$, which takes into account signs of crossings changed to get a slice knot and which is bounded above by the slicing number and below by the slice genus. We exhibit an infinite family of knots $ K_n$ with slice genus $ n$ and Livingston invariant greater than $ n$. Our bounds are based on restrictions (using Donaldson's diagonalisation theorem or Heegaard Floer homology) on the intersection forms of four-manifolds bounded by the double branched cover of a knot.


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Additional Information

Brendan Owens
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
Address at time of publication: Department of Mathematics, University of Glasgow, Glasgow, G12 8QW, United Kingdom

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04904-6
PII: S 0002-9947(09)04904-6
Received by editor(s): April 11, 2008
Published electronically: August 13, 2009
Additional Notes: The author was supported in part by NSF grant DMS-0604876.
Article copyright: © Copyright 2009 American Mathematical Society