Given integers \(H,p,q\) and a real number \(\phi \), consider the loop \(\gamma\) in 3-dimensional space parametrized by
\[x(\theta) = (2+\sin(q\theta))\cos(H \theta)\] \[y(\theta) = (2+\sin(q \theta))\sin(H \theta) \] \[z(\theta) = \cos(p (\theta+\phi)) \]
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 of the unit disk
\[ \nabla^2 F = 0 \qquad F|_{\partial \mathbb{D}^2} = f,\] one obtains harmonic coordinate functions for a singular map of a disk into the 3-sphere.
Displayed right is a plot of this singular disk.
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.