
## 1. Filtrations

1. ### Reversal and the bipolar filtration

#### 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

#### 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

#### Problem 1.3.

[Min Hoon Kim] Are all 2-torsion knots 0-bipolar?
• ### Characterization of 0-bipolar knots

#### 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

#### 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.

#### 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.