Let P be a finite set of points on the unit sphere $S^2$ such that

  • for every $p\in P$, there exists a closed curve $\gamma_p \subset S^2$ which has a self intersection at $p$ and passes through $-p$. Moreover, every plane passing through $p$ and the origin intersect $\gamma_p$ in at most 2 points, excluding $-p$ (see the graph below by Neil Strickland).

  • At the self intersection point $p$, $\gamma_p$ is orthogonal to itself.

  • $\gamma_{p_1} \cap \gamma_{p_2} \subset P$ for every $p_1,p_2 \in P$

  • If $A_i \subset S^2$ be such that $\partial A_i \subset \gamma_{p_i}$, $p_i\in P$, i=1,2, then neither $A_1 \subset A_2$ nor $A_2 \subset A_1$.

  • $P \subset \gamma_p$ for all $p\in P$.

Can we prove that $P$ has at most one element? If not, what is the upper bound on the number of elements of $P$?

  • 2
    $\begingroup$ Please write couple of words on motivation. $\endgroup$ – Anton Petrunin Sep 5 '17 at 20:51
  • $\begingroup$ Is it OK to assume that $\gamma_p$ is smooth? $\endgroup$ – Anton Petrunin Sep 5 '17 at 23:01
  • $\begingroup$ Yes, $\gamma_p$ is indeed smooth. $\endgroup$ – User4966 Sep 5 '17 at 23:24
  • $\begingroup$ I find it hard to imagine even two of these curves satisfying all assumptions. Could you draw a picture with two or three curves, please? $\endgroup$ – Sebastian Goette Sep 6 '17 at 16:26
  • $\begingroup$ Yes, it's difficult to draw a picture. Indeed if P has more than one element , then it should have at least 4 element. $\endgroup$ – User4966 Sep 7 '17 at 5:07

This is just a comment, really. One can write a nice formula for a family of curves of the indicated type, as follows. Let $p$, $q$ and $r$ be an orthonormal basis of $\mathbb{R}^3$, and suppose that $a>0$. Put $$ C = \{x\in S^2: (q.x)\,(r.x)=a((q-r).x)(1 - p.x)\} $$ enter image description here

This can be parametrised as $$ \gamma(t) = \frac{(a^2(c-s)^2-s^2c^2)p+2asc(c-s)(cq+sr)}{a^2(c-s)^2+s^2c^2}, $$ where $s=\sin(t/2)$ and $c=\cos(t/2)$. This passes through $p$ with derivative $q/a$ at $t=0$ and through $p$ again with derivative $r/a$ at $t=\pi$. It also passes through $-p$ with derivative $-2a(q+r)$ at $t=\pi/2$. Maple code is as follows:

g := unapply(subs({s = sin(t/2),c=cos(t/2)},((a^2*( c-s)^2-s^2*c^2) *~ 
 [1,0,0] +~ (2*a*( c-s)*( s*c)) *~ [0, c, s])/~(a^2*( c-s)^2+s^2*c^2)),t);

Obviously this does not answer the question, but you might gain some intuition by plotting curves of this type.


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