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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.