A373325 Number of semi-infinite curves of the plane with n simple, transverse self-intersections and no other self-intersections, up to an orientation-preserving homeomorphism.
1, 2, 10, 66, 498, 4072, 35144, 315352, 2914074, 27553880, 265387528, 2595131328
Offset: 0
Examples
Curves without self-intersection are equivalent; one might for instance take the half-line y <= 0 as their representative; so a(0) = 1. To get a curve with n+1 self-intersections, one can start from a curve with n self-intersections; identify the cycle of oriented edges that directly surrounds the finite extremity of the curve; choose an edge from that cycle and extend the curve so that it crosses that edge. When "outside" it might help visualization to imagine that a noncrossable oriented edge "at infinity" closes the cycle. Thus, for a transition between 0 and 1 self-intersection, the choice is between making a loop that turns left and making a loop that turns right; so a(1) = 2. See provided illustration for n=0..3 in section 'Links'.
Links
- Luc Rousseau, Illustration for n=0..3
- Luc Rousseau, A Prolog program.
