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

## 3. Structure and operators

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

#### 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].

#### Problem 3.2.

[Shelly Harvey] Find a knot $K$ with $0< g_4^{st}(K)< 1/2$.
• ### Stable 4-genus and 2-torsion

#### 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

#### 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?

#### 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.