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5. Miscellaneous

    1. Rational sliceness


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      Problem 5.1.

      [Jae Choon Cha] Is every knot in $S^3$ which is slice in some rational homology ball actually slice in some $\mathbb{Z}[1/2]$-homology ball?

        • Difficult sliceness problems


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          Problem 5.2.

          Are any of the following slice?
          1. The $(2,1)$ cable of the figure-eight knot.
          2. The positive Whitehead double of any left-handed torus knot.
          3. The positive Whitehead double of the figure-eight knot.
          4. The $(+,+,-)$ Whitehead double of the Borromean rings.
            1. Remark. It is known that $C_{2,1}(4_1)$ is rationally slice but is not ribbon.
                • Remark. [Jae Choon Cha] The problem of sliceness of $C_{2,1}(4_1)$ is important for the classification of the $L^2$-acyclic bordism group.

                    • Attacks on slice-ribbon


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                      Problem 5.3.

                      [Clayton McDonald] Find new ways to build knots which are by construction slice but not by construction ribbon.
                        1. Remark. Gompf-Scharlemann-Thompson give some examples, but their construction produces intrinsically homotopy ribbon knots.

                          Cochran-Davis give some examples as well.
                            • Remark. [Adam Levine] One might also want to switch perspective to ribbon links, where there is a Jones polynomial obstruction due to Eisermann.

                                • Understanding topological slice discs


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                                  Problem 5.4.

                                  [Maggie Miller] Explicitly describe some non-smooth topologically slice disc.
                                      Potential approach: one might try for some sort of infinite band sum of an infinite component unlink.

                                    • Topological ribbon concordance

                                          Gordon asked whether two knots which are mutually ribbon concordant must be isotopic. This problem offers a topological version.

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                                      Problem 5.5.

                                      [Mark Powell] Define knots $K$ and $J$ to be topologically homotopy ribbon concordant, and write $K \leq_{thrc} J$, if $K$ and $J$ are topologically concordant via an annulus $A$ such that the inclusion-induced map $\pi_1(X_K) \to \pi_1(X_A)$ is a surjection and the inclusion-induced map $\pi_1(X_J) \to \pi_1(X_A)$ is an injection. Is $\leq_{thrc}$ a partial order on the collection of knots in $S^3$?

                                        • Grope metric


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                                          Problem 5.6.

                                          [Shelly Harvey] Understand the grope metric $d_1$ defined by Cochran-Harvey-Powell [MR3665407].
                                              A closely related question is whether there exists a non smoothly slice knot which bounds an infinite height symmetric grope where all surfaces have genus 1, and whether such a knot must be topologically slice.

                                            • Metrics on $\mathcal{T}$


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                                              Problem 5.7.

                                              [Shelly Harvey] Construct an interesting (i.e. non-discrete, perhaps nice with respect to the bipolar filtration) metric on the set of topologically slice knots.

                                                • Round Handle Problem


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                                                  Problem 5.8.

                                                  [Mark Powell] Given a linking number 0 link $L$, construct a 4-manifold $W$ by attaching a round handle to $B^4$ along each component of $L$. Must $L$ be topologically slice in $W$?
                                                    1. Remark. [Mark Powell] If both topological surgery and the s-cobordism theorem for 4-manifolds hold, then the answer is ‘Yes’, so it would be very interesting to prove ’No’.

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