Experimental Mathematics

Self-Intersection Numbers of Curves on the Punctured Torus

Moira Chas and Anthony Phillips

Full-text: Open access


On the punctured torus the number of essential self-intersections of a homotopy class of closed curves is bounded (sharply) by a quadratic function of its combinatorial length (the number of letters required for its minimal description in terms of the two generators of the fundamental group and their inverses). We show that if a homotopy class has combinatorial length $L$, then its number of essential self-intersections is bounded by $(L-2)^{2}/4$ if $L$ is even, and $(L-1)(L-3)/4$ if $L$ is odd. The classes attaining this bound can be explicitly described in terms of the generators; there are $(L-2)^2+ 4$ of them if $L$ is even, and $2(L-1)(L-3)+8$ if $L$ is odd. Similar descriptions and counts are given for classes with self-intersection number equal to one less than the bound. Proofs use both combinatorial calculations and topological operations on representative curves.

Computer-generated data are tabulated by counting for each nonnegative integer how many length-$L$ classes have that self-intersection number, for each length $L$ less than or equal to $13$. Such experiments led to the results above. Experimental data are also presented for the pair-of-pants surface.

Article information

Experiment. Math., Volume 19, Issue 2 (2010), 129-148.

First available in Project Euclid: 17 June 2010

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M05: Fundamental group, presentations, free differential calculus
Secondary: 57N50: $S^{n-1}\subset E^n$, Schoenflies problem 30F99: None of the above, but in this section

Punctured torus free homotopy classes of curves self-intersection


Chas, Moira; Phillips, Anthony. Self-Intersection Numbers of Curves on the Punctured Torus. Experiment. Math. 19 (2010), no. 2, 129--148. https://projecteuclid.org/euclid.em/1276784785

Export citation