%0 Generic %1 Chas2009 %A Chas, Moira %A Phillips, Anthony %D 2009 %K mapping surfaces %T Self-intersection numbers of curves on the punctured torus %U http://arxiv.org/abs/0901.2974 %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.

@misc{Chas2009, abstract = { 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. }, added-at = {2010-11-01T12:36:28.000+0100}, author = {Chas, Moira and Phillips, Anthony}, biburl = {https://www.bibsonomy.org/bibtex/2ab73a163727ac64ac5e0b977ccc6206b/uludag}, description = {Self-intersection numbers of curves on the punctured torus}, interhash = {3bd2a8f747484e504a5c20ad516e8f46}, intrahash = {ab73a163727ac64ac5e0b977ccc6206b}, keywords = {mapping surfaces}, note = {cite arxiv:0901.2974 Comment: 39 pages, 8 figures}, timestamp = {2010-11-01T12:36:28.000+0100}, title = {Self-intersection numbers of curves on the punctured torus}, url = {http://arxiv.org/abs/0901.2974}, year = 2009 }