I'm looking for a parameterization $(x_1(u,v),x_2(u,v),x_3(u,v),x_4(u,v))$ of a knotted sphere in $\mathbb R^4$. How might one go about finding such a parameterization?

  • 1
    $\begingroup$ Which knotted surface? $\endgroup$ – Igor Rivin Dec 3 '15 at 11:23
  • $\begingroup$ @IgorRivin Yeah, I left that deliberately open. I'm looking for a parameterization of any relatively basic knotted surface, that I can use to produce a visualization of the intersection of the surface with a family of 3d affine hyperplanes, and seeing how the intersections evolve over time as classical links. $\endgroup$ – Reinstate Monica Dec 3 '15 at 16:38
  • $\begingroup$ The question is: what do you mean by "knotted"? Which is the same question as: what do you mean by UNknotted? $\endgroup$ – Igor Rivin Dec 3 '15 at 16:44
  • $\begingroup$ @IgorRivin I suppose I should restrict to spheres. Then take unkotted to be ambient isotopic to an embedding that lies in $\mathbb R^3$ $\endgroup$ – Reinstate Monica Dec 3 '15 at 16:59
  • $\begingroup$ I'd understand if this question was considered too basic for this site, but I don't see how it is at all unclear what is being asked for. $\endgroup$ – PVAL Dec 3 '15 at 19:53

If you know how to parameterize a nontrivial knot in $R^3$ (say, the trefoil knot), you can use Artin's "spinning" construction to parameterize a nontrivial knot in $R^4$ via map $S^2\to R^4$ written in spherical coordinates on $S^2$. Take, say, the trefoil $T$ in $R^3$ parameterized by some smooth periodic function $f(\theta), 0\le \theta\le a$ where $a$ is the period. Let $L$ denote the straight line in $R^3$ through the points $f(0), f(c)$, $c=a- \epsilon$ where $\epsilon>0$ is sufficiently small (if you want to know how small is small, ask me and I can explain). I will choose the coordinates $x_1, x_2, x_3$ so that $L$ is the $x_3$-axis. Let $R_\phi$ denote the rotation by the angle $\phi$ in $R^4$ around the plane $x_1=x_2=0$ (thus, fixing $L$). Now, consider the map $F: [0, c]\times [0, 2\pi]\to R^4$, given by $$F(\theta, \phi) = R_\phi\circ f(\theta)$$ Its image is a knotted 2-sphere $\Sigma$ in $R^4$. It is smooth away from the points $f(0), f(c)$. If this bothers you, it can be fixed (explicitly). Now, rescale the interval $[0, c]$ to $[0, \pi]$ via the map $s: t\mapsto \frac{\pi}{c}t$. Then the map $$G= F\circ (s^{-1} \times Id): [0,\pi]\times [0,2\pi]\to R^4$$ parameterizes the same surface $\Sigma$ as before. Lastly, use the spherical coordinates on $S^2$ to convert the map $G$ to a homeomorphism $S^2\to \Sigma$.

This, I think, is as an explicit construction as it can be (provided you are not bothered by smallness issue for $\epsilon$).

Edit: Upon your request, smallness of $\epsilon$ is defined as follows: There exists a round ball $B$ (or a half-space) in $R^3$ with points $p=f(a-\epsilon), q=f(a)$ on its boundary such that the arc $\alpha=f([a-\epsilon, a])$ is contained in $B$ and is unknotted in $B$, i.e., is isotopic to the line segment $pq$ in $B$ via an isotopy fixing the boundary sphere of $B$. (In practical terms, you can think of the unknottedness property of $\alpha$ as assuming that $\alpha$ is a graph of a function on the segment $pq$, with values in the plane orthogonal to $pq$.)

See for instance here for details.

| cite | improve this answer | |
  • $\begingroup$ Thanks for your answer. How small is sufficiently small? $\endgroup$ – Reinstate Monica Dec 5 '15 at 0:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.