This is a plot of a singular disk bounded by a Lissajous-toric knot.
Given integers \(H,p,q\) and a real number \(\phi \), consider the loop defined in coordinates 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 \(\gamma\) parametrizes 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. Seen right is a plot of this singular disk.
Remark: unbounded disks are artificially truncated.