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\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

3. Structure and operators

    1. 2-torsion in $\mathcal{C}$


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      Problem 3.1.

      [Jen Hom] Can one detect whether $K$ is concordant to a negative amphichiral knot?
        1. Remark. This is related to Gordon’s conjecture that all 2-torsion elements of $\mathcal{C}$ are concordant to negative amphichiral knots.

            • Small stable 4-genera

                  The stable 4-genus of a knot $K$, in either category, is defined as $g_4^{st}(K)= \lim_{n \to \infty} \frac{ g_4(nK)}{n}$, see Livingston [MR2745668].

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              Problem 3.2.

              [Shelly Harvey] Find a knot $K$ with $0< g_4^{st}(K)< 1/2$.

                • Stable 4-genus and 2-torsion


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                  Problem 3.3.

                  Find a knot $K$ which has infinite order in $\mathcal{C}$ and yet which has $g_4^{st}(K)=0$.

                    • Injectivity of winding number 0 satellite operators


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                      Problem 3.4.

                      [Shelly Harvey] Is there a pattern $P$ with winding number 0 such that $P(K)$ is slice if and only if $K$ is slice, in either category?
                        1. Remark. [Allison Miller] This is open for patterns of any winding number besides $\pm1$.

                            • Does the homotopy class of the infection curve determine the action of a pattern on topological concordance?


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                              Problem 3.5.

                              [Arunima Ray] Does the Mazur pattern act by the identity on the topological concordance group? More generally, if $\eta_1$ and $\eta_2$ are two homotopic curves in the complement of a slice knot $R$, must the induced maps $R_{\eta_1}$ and $R_{\eta_2}$ on the topological concordance group agree?
                                1. Remark. This is certainly false in the smooth category. See for example the Whitehead doubling operator.
                                    • Remark. [Allison Miller] If ‘No’, then it follows from work of Yasui [arXiv:1505.02551] that the trace Akbulut-Kirby conjecture is false in the topological category.

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