Algebraic & Geometric Topology

Positive links

Tim D Cochran and Eamonn Tweedy

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Given a link LS3, 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.

Article information

Algebr. Geom. Topol., Volume 14, Number 4 (2014), 2259-2298.

Received: 12 April 2013
Revised: 12 December 2013
Accepted: 8 January 2014
First available in Project Euclid: 19 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M27: Invariants of knots and 3-manifolds 57N70: Cobordism and concordance

concordance slice link $4$–manifold


Cochran, Tim D; Tweedy, Eamonn. Positive links. Algebr. Geom. Topol. 14 (2014), no. 4, 2259--2298. doi:10.2140/agt.2014.14.2259.

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