%0 Journal Article
%T Self-intersection numbers of curves on the punctured torus
%A Moira Chas
%A Anthony Phillips
%J Mathematics
%D 2009
%I arXiv
%X The minimum number of self-intersection points for members of a free homotopy class of curves on the punctured torus is bounded above in terms of the number L of letters required for a minimal description of the class in terms of the generators of the fundamental group and their inverses: it is less than or equal to (L-2)^2/4 if L is even, and (L-1)(L-3)/4 if L is odd. The classes attaining this bound are 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 totals are given for classes with self-intersection number equal to one less than the maximum. Proofs use both combinatorial calculations and topological operations on representative curves. Computer-generated data are tabulated counting, for each non-negative integer, how many length-L classes have that self-intersection number, for each length L less than or equal to 12. Experimental data are also presented for the pair-of-pants surface.
%U http://arxiv.org/abs/0901.2974v1