Warning: Large parameter values may use a lot of memory
Configurations where some (or all) of \(H,p,q\) are negative will produce unbounded surfaces which we
artificially truncate 10 units from the center.
Selected configurations
| H |
p |
q |
| 9 |
16 |
41 |
| 7 |
10 |
31 |
| 7 |
15 |
44 |
| 11 |
15 |
34 |
| -10 |
33 |
9 |
What am I looking at?
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 parametrized loop \(\gamma : S^1 \rightarrow \mathbb{R}^3\) defined via the coordinate functions
\[x(e^{i\theta}) = (2+\sin(q\theta))\cos(H \theta)\] \[y(e^{i\theta}) = (2+\sin(q \theta))\sin(H \theta) \] \[z(e^{i\theta}) = \cos(p (\theta+\phi)) \]
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 the punctured disk \(D : \mathbb{D}^2 \setminus \{0\} \rightarrow \mathbb{R}^3 \).
When all parameters are positive, the domain may be extended to the entire disk. When negative parameters are present, the map can be extended to the full disk by appending point at infinity to \(\mathbb{R}^3\).