5. Miscellaneous

Rational sliceness
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
Problem 5.2.
Are any of the following slice? The $(2,1)$ cable of the figureeight knot.
 The positive Whitehead double of any lefthanded torus knot.
 The positive Whitehead double of the figureeight knot.
 The $(+,+,)$ Whitehead double of the Borromean rings.

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 sliceribbon
Problem 5.3.
[Clayton McDonald] Find new ways to build knots which are by construction slice but not by construction ribbon.
Remark. GompfScharlemannThompson give some examples, but their construction produces intrinsically homotopy ribbon knots.
CochranDavis 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
Problem 5.4.
[Maggie Miller] Explicitly describe some nonsmooth topologically slice disc. 
Topological ribbon concordance
Gordon asked whether two knots which are mutually ribbon concordant must be isotopic. This problem offers a topological version.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 inclusioninduced map $\pi_1(X_K) \to \pi_1(X_A)$ is a surjection and the inclusioninduced 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
Problem 5.6.
[Shelly Harvey] Understand the grope metric $d_1$ defined by CochranHarveyPowell [MR3665407]. 
Metrics on $\mathcal{T}$
Problem 5.7.
[Shelly Harvey] Construct an interesting (i.e. nondiscrete, perhaps nice with respect to the bipolar filtration) metric on the set of topologically slice knots. 
Round Handle Problem
Problem 5.8.
[Mark Powell] Given a linking number 0 link $L$, construct a 4manifold $W$ by attaching a round handle to $B^4$ along each component of $L$. Must $L$ be topologically slice in $W$?
Remark. [Mark Powell] If both topological surgery and the scobordism theorem for 4manifolds 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.