Knot traces and concordance

Abstract

We give a method for constructing many pairs of distinct knots $K_0$ and $K_1$ such that the two 4-manifolds obtained by attaching a 2-handle to $B^4$ along $K_i$ with framing zero are diffeomorphic. We use the d-invariants of Heegaard Floer homology to obstruct the smooth concordance of some of these $K_0$ and $K_1$, thereby disproving a conjecture of Abe in [Abe16]. As a consequence, we obtain a proof that there exist patterns $P$ in solid tori such that $P(K)$ is not always concordant to $P(U)\#K$ and yet whose action on the smooth concordance group is invertible.

17 pages, 15 figures. This version of the paper has been accepted by the Journal of Topology for publication

References in corpus (5)

• Classification of trisections and the Generalized Property R Conjecture
• Corks, exotic 4-manifolds and knot concordance
• Waldhausen's Theorem
• Shake slice and shake concordant knots
• Annulus twist and diffeomorphic 4-manifolds II

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• Shake genus and slice genus
• The trace embedding lemma and spinelessness
• Notes on constructions of knots with the same trace
• On annulus presentations, dualizable patterns and RGB-diagrams
• Stein traces and characterizing slopes
• Ribbon disks with the same exterior
• Knots with infinitely many non-characterizing slopes
• From zero surgeries to candidates for exotic definite four-manifolds
• The Complexity of Shake Slice Knots
• Action of the Mazur pattern up to topological concordance