Immersed disks, slicing numbers and concordance unknotting numbers

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.Comment: 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. Geo

Dehn surgeries and negative-definite four-manifolds

Given a knot <i>K</i> in the three-sphere, we address the question: Which Dehn surgeries on <i>K</i> bound negative-definite four-manifolds? We show that the answer depends on a number <i>m(K)</i>, which is a smooth concordance invariant. We study the properties of this invariant and compute it for torus knots

BOOK REVIEWS

Reviews of; Historical and Cultural Perspectives on Slovenian Migration by Drnovšek Marjan (ed.) (2007) Ljubljana: Institute for Slovenian Emigration Studies, 204 pp. (ISBN 978-961-254-043-2); Migration, Gender and National Identity: Spanish Migrant Women in London Ana Bravo Moreno (2006) Peter Lang, Oxford, Bern, Berlin, Bruxelles, Frankfurt am Main, New York Wien (ISBN 3-03910-156-0). Key words Slovenian migration; Cultural Perspectives; Migration Policy; Migration and Gender; Identity; Spanish migrants; London

Smooth concordance of links topologically concordant to the Hopf link

It was shown by Jim Davis that a 2-component link with Alexander polynomial one is topologically concordant to the Hopf link. In this paper, we show that there is a 2-component link with Alexander polynomial one that has unknotted components and is not smoothly concordant to the Hopf link, answering a question of Jim Davis. We construct infinitely many concordance classes of such links, and show that they have the stronger property of not being smoothly concordant to the Hopf link with knots tied in the components.Comment: 8 pages, 5 figure