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1. Filtrations

    1. Reversal and the bipolar filtration


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      Problem 1.1.

      [Maggie Miller] For each $n \in \mathbb{N}$, find an $n$-bipolar topologically slice knot $K$ such that $K \# -K^r$ is not smoothly slice.

        • 4-genus and the bipolar filtration


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          Problem 1.2.

          [Jae Choon Cha] For $n \in \mathbb{N}$, are there knots in $\mathcal{T}_n$ with arbitrarily large smooth 4-genus?

            • Torsion and the bipolar filtration


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              Problem 1.3.

              [Min Hoon Kim] Are all 2-torsion knots 0-bipolar?

                • Characterization of 0-bipolar knots


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                  Problem 1.4.

                  [Shelly Harvey] Can one characterize 0-positive, 0-negative, or 0-bipolar knots, either via the vanishing of invariants or geometrically?
                    1. Remark. [Arunima Ray] In the solvable filtration, a knot $K$ is 0-solvable if and only if its Arf invariant vanishes.

                        • Bipolar quotients


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                          Problem 1.5.

                          Does $\mathcal{T}_n/ \mathcal{T}_{n+1}$ contain a $\mathbb{Z}^{\infty}$-summand? What about a $(\mathbb{Z}/2\mathbb{Z})^{\infty}$ subgroup?
                              In fact, it is unknown in general even if $\mathcal{T}_n/ \mathcal{T}_{n+1}$ contains a $\mathbb{Z}/ 2\mathbb{Z}$ subgroup. Work of Cha-Kim [MR3228458] shows that $\mathcal{T}_n/ \mathcal{T}_{n+1}$ contains a $\mathbb{Z}^{\infty}$- subgroup for all $n \in \mathbb{N}$.

                            • Highly solvable knots with large 4-genera.


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                              Problem 1.6.

                              [Jae Choon Cha] For arbitrary $n>2$ and $g>0$ prove that there exist $n$-solvable knots $K$ such that the topological 4-genus of $K$ is strictly more than $g$.
                                  The $n=0$ case follows from Tristram-Levine signatures, the $n=1$ case from Casson-Gordon signatures, and the $n=2$ case from work of Cha-Miller-Powell [arXiv:1901.02060].

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