Given integers \(H,p,q\) and a real number \(\phi \), consider the loop \(\gamma\) in 3-dimensional space parametrized by
\[x(e^{i\theta}) = (2+\sin(q\theta))\cos(H \theta)\] \[y(e^{i\theta}) = (2+\sin(q \theta))\sin(H \theta) \] \[z(e^{i\theta}) = \cos(p (\theta+\phi)) \]
When \(\gcd(H,p)=\gcd(H,q)=1\) and \(\phi\) is chosen so that the image of \(\gamma\) does not self-intersect, the curve \(\gamma\) is the Lissajous-toric knot \(K(H,p,q,\phi)\).
Using the coordinate functions above as boundary conditions for the Dirichlet problem
\[ \nabla^2 F = 0 \qquad F|_{\partial \mathbb{D}^2} = f,\] one obtains coordinate functions for a singular map of a disk into the 3-sphere.
By construction, all coordinate functions are harmonic.
Question for mathematicians
If \(\gcd(p,q)=1\) then \(J:=K(H,p,q,\phi)\) bounds a
smooth ribbon disk in the 4-dimensional ball.
For such \(J\), is its Dirichlet disk (or a perturbation thereof) smoothly
isotopic through immersions
to an embedded disk in the 4-ball?
If not, are there any smooth 4-manifolds where the above holds?
Comments on the plot
Color gradient is determined by normal vectors to the surface.
One may use the control panel in the top-right to change the
knot and the opacity of the surface.
Unbounded disks are artificially truncated.