3 Acknowledged error.

Hey Joel, long time etc. It looks to me like blowing down your knotted $S^2$ will only produce a homology 4-sphere. And one could presumably produce examples by taking some known 2-knot in $S^4$ and connect-summing it with the line in $\mathbb{CP}^2$, distinguishing the resulting 2-knots in $\mathbb{CP}^2$ from the line via $\pi_1$ of their complements.

[EDIT: I fell into Joel's heffalump trap. Still, at least there's company down here...]

You could rephrase the question (with a bit of help from Gromov) as asking whether a 2-knot in $\mathbb{CP}^2$ with self-intersection $1$ and simply connected complement is isotopic to a symplectic sphere. You could invoke Taubes too, and see that, to produce a diffeo with the line, it's enough to extend a symplectic form on the image of $S^2$ to one on $\mathbb{CP}^2$. Well, the complement of a neighbourhood of $S^2$ is then a homotopy 4-ball, bounding $S^3$ with its usual contact structure, and the goal is to build a symplectic form which is a convex filling of the contact boundary... Yep, that's probably an open problem.

2 corrected "concave" to "convex"

Hey Joel, long time etc. It looks to me like blowing down your knotted $S^2$ will only produce a homology 4-sphere. And one could presumably produce examples by taking some known 2-knot in $S^4$ and connect-summing it with the line in $\mathbb{CP}^2$, distinguishing the resulting 2-knots in $\mathbb{CP}^2$ from the line via $\pi_1$ of their complements.

You could rephrase the question (with a bit of help from Gromov) as asking whether a 2-knot in $\mathbb{CP}^2$ with self-intersection $1$ and simply connected complement is isotopic to a symplectic sphere. You could invoke Taubes too, and see that, to produce a diffeo with the line, it's enough to extend a symplectic form on the image of $S^2$ to one on $\mathbb{CP}^2$. Well, the complement of a neighbourhood of $S^2$ is then a homotopy 4-ball, bounding $S^3$ with its usual contact structure, and the goal is to build a symplectic form which is a concave convex filling of the contact boundary... Yep, that's probably an open problem.

1

Hey Joel, long time etc. It looks to me like blowing down your knotted $S^2$ will only produce a homology 4-sphere. And one could presumably produce examples by taking some known 2-knot in $S^4$ and connect-summing it with the line in $\mathbb{CP}^2$, distinguishing the resulting 2-knots in $\mathbb{CP}^2$ from the line via $\pi_1$ of their complements.

You could rephrase the question (with a bit of help from Gromov) as asking whether a 2-knot in $\mathbb{CP}^2$ with self-intersection $1$ and simply connected complement is isotopic to a symplectic sphere. You could invoke Taubes too, and see that, to produce a diffeo with the line, it's enough to extend a symplectic form on the image of $S^2$ to one on $\mathbb{CP}^2$. Well, the complement of a neighbourhood of $S^2$ is then a homotopy 4-ball, bounding $S^3$ with its usual contact structure, and the goal is to build a symplectic form which is a concave filling of the contact boundary... Yep, that's probably an open problem.