Volume 14, issue 4 (2014)

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Positive links

Tim D Cochran and Eamonn Tweedy

Algebraic & Geometric Topology 14 (2014) 2259–2298

Given a link L S3, we ask whether the components of L bound disjoint, nullhomologous disks properly embedded in a simply connected positive-definite smooth 4–manifold; the knot case has been studied extensively by Cochran, Harvey and Horn. Such a 4–manifold is necessarily homeomorphic to a (punctured) #kP(2). We characterize all links that are slice in a (punctured) #kP(2) in terms of ribbon moves and an operation which we call adding a generalized positive crossing. We find obstructions in the form of the Levine–Tristram signature function, the signs of the first author’s generalized Sato–Levine invariants, and certain Milnor’s invariants. We show that the signs of coefficients of the Conway polynomial obstruct a 2–component link from being slice in a single punctured P(2) and conjecture these are obstructions in general. These results have applications to the question of when a 3–manifold bounds a 4–manifold whose intersection form is that of some #kP(2). For example, we show that any homology 3–sphere is cobordant, via a smooth positive-definite manifold, to a connected sum of surgeries on knots in S3.

concordance, slice link, $4$–manifold
Mathematical Subject Classification 2010
Primary: 57M25
Secondary: 57M27, 57N70
Received: 12 April 2013
Revised: 12 December 2013
Accepted: 8 January 2014
Published: 28 August 2014
Tim D Cochran
Department of Mathematics MS-136
Rice University
PO Box 1892
Houston, TX 77251-1892
Eamonn Tweedy
Department of Mathematics MS-136
Rice University
PO Box 1892
Houston, TX 77251-1892