Following the disproof by Livingston, Kirk and Yasui of the conjecture of Akbulut-Kirby that a homeomorphism between spaces obtained by Dehn \(0\)-surgery on distinct knots \(K\) and \(K'\) implies the knots are smoothly concordant, the conjecture was modified by Abe as follows: Suppose the knots \(K\) and \(K'\) lead to the \(4\)-manifolds \(X_{0}(K)\), \(X_{0}(K')\) when a \(0\)-framed \(2\)-handle is added to \(B^4\) by attaching the handle along a neighborhood of \(K\), \(K'\) respectively. Suppose the resulting \(4\)-manifolds are diffeomorphic. Then up to reversing orientation of either knot, the knots are smoothly concordant.
The main theorem of the present work disproves this conjecture as well. The authors construct an infinite family of pairs of knots which contradict the conjecture. The construction of the pairs utilizes Dehn twists of “dualizable” patterns (knots) in a solid torus.
A pattern \(P\) acts on the concordance group of knots by sending a representative \(K\) of a concordance class to \(P(K)\), a representative of the concordance class containing the knot obtained from the pattern \(P\) in a solid torus knotted like \(K\). Thus \(P(U) = P\) for the unknot \(U\). A pattern \(P\) is called invertible if there is a pattern \(Q\) such that \(P(Q(K))\) is concordant to both \(K\) and \(Q(P(K))\) for any \(K\). The authors show that there are invertible patterns which do not act by connected sum on the smooth concordance group.
The patterns in the proof of the main theorem are distinguished by their Alexander polynomials, the diffeomorphic character of the \(4\)-manifolds in question by theorems on dualizable patterns, and the non-concordance of the knots by establishing the non-concordance of the knots’ \(2\)-sheeted branched coverings. This latter non-concordance follows from examination of the \(d\)-invariants of P. Ozsváth and Z. Szabó [Adv. Math. 173, No. 2, 179–261 (2003; Zbl 1025.57016)].
The authors relate their analyses of dualizable patterns to annulus presentations, and add other observations about their family of counterexamples.