Volume 23, issue 5 (2019)

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Shake genus and slice genus

Lisa Piccirillo

Geometry & Topology 23 (2019) 2665–2684

An important difference between high-dimensional smooth manifolds and smooth 4–manifolds that in a 4–manifold it is not always possible to represent every middle-dimensional homology class with a smoothly embedded sphere. This is true even among the simplest 4–manifolds: X0(K) obtained by attaching an 0–framed 2–handle to the 4–ball along a knot K in S3. The 0–shake genus of K records the minimal genus among all smooth embedded surfaces representing a generator of the second homology of X0(K) and is clearly bounded above by the slice genus of K. We prove that slice genus is not an invariant of X0(K), and thereby provide infinitely many examples of knots with 0–shake genus strictly less than slice genus. This resolves Problem 1.41 of Kirby’s 1997 problem list. As corollaries we show that Rasmussen’s s invariant is not a 0–trace invariant and we give examples, via the satellite operation, of bijective maps on the smooth concordance group which fix the identity but do not preserve slice genus. These corollaries resolve some questions from a conference at the Max Planck Institute, Bonn (2016).

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knot traces, shake genus, slice genus
Mathematical Subject Classification 2010
Primary: 57M25, 57R65
Received: 9 May 2018
Revised: 17 September 2018
Accepted: 1 January 2019
Published: 13 October 2019
Proposed: Rob Kirby
Seconded: Ciprian Manolescu, Walter Neumann
Lisa Piccirillo
Department of Mathematics
University of Texas
Austin, TX
United States