Loading Web-Font TeX/Math/Italic
| Register
\newcommand{\Cat}{{\rm Cat}} \newcommand{\A}{\mathcal A} \newcommand{\freestar}{ \framebox[7pt]{$\star$} }

4. Determining concordance

    1. Genus g traces and concordance


      Add remark

      Problem 4.1.

      [Lisa Piccirillo] For a knot K in S^3, let X^g(K) denote the 4-ball union \Sigma_g \times D^2, where \Sigma_g is a genus g surface with one boundary component, attached to B^4 along a 0-framed neighborhood of K. Are there nonconcordant knots K and J such that \partial X^g(K) and \partial X^g(J) are integrally homology cobordant?
        1. Remark. [Lisa Piccirillo] Note that homeomorphism of \partial X^g(K) and \partial X^g(J) is enough to imply that the knots K and J are isotopic.

            • Branched covers and concordance


              Add remark

              Problem 4.2.

              Suppose K and J are knots in S^3 such that for every prime power q, the qth cyclic branched covers \Sigma_q(K) and \Sigma_q(J) are rationally homology cobordant. Must K and J be concordant?

                • Surgeries and concordance


                  Add remark

                  Problem 4.3.

                  Suppose K and J are knots in S^3 such that for all p/q \in \mathbb{Q}, the Dehn surgeries S^3_{p/q}(K) and S^3_{p/q}(J) are integer homology cobordant. Must K and J be concordant?

                    • 0-surgeries and concordance


                      Add remark

                      Problem 4.4.

                      [Adam Levine] Suppose K and J are knots in S^3 such that there is an integer homology cobordism W between S^3_0(K) and S^3_0(J) in which the positively oriented meridians \mu_K and \mu_J are freely homotopic. Must K and J be concordant?
                        1. Remark. Under the assumption that \mu_K and \mu_J are actually concordant in W, it follows that K and J are concordant (in the topological category) or exotically concordant (in the smooth category.)
                            • Remark. [Arunima Ray] This question for links was addressed by Cha-Powell [arXiv:1309.5051], who showed that the answer was ’no’.

                                  Cite this as: AimPL: Smooth concordance classes of topologically slice knots, available at http://aimpl.org/concordsliceknot.