%0 Generic %1 Chas2010 %A Chas, Moira %A Phillips, Anthony %D 2010 %K geodesic surfaces %T Self-intersection numbers of curves in the doubly-punctured plane %U http://arxiv.org/abs/1001.4568 %X We address the problem of computing bounds for the self-intersection number (the minimum number of self-intersection points) of members of a free homotopy class of curves in the doubly-punctured plane as a function of their combinatorial length L; this is the number of letters required for a minimal description of the class in terms of the standard generators of the fundamental group and their inverses. We prove that the self-intersection number is bounded above by L^2/4 + L/2 - 1, and that when L is even, this bound is sharp; in that case there are exactly four distinct classes attaining that bound. When L is odd, we establish a smaller, conjectured upper bound ((L^2 - 1)/4)) in certain cases; and there we show it is sharp. Furthermore, for the doubly-punctured plane, these self-intersection numbers are bounded below, by L/2 - 1 if L is even, (L - 1)/2 if L is odd; these bounds are sharp.

@misc{Chas2010, abstract = { We address the problem of computing bounds for the self-intersection number (the minimum number of self-intersection points) of members of a free homotopy class of curves in the doubly-punctured plane as a function of their combinatorial length L; this is the number of letters required for a minimal description of the class in terms of the standard generators of the fundamental group and their inverses. We prove that the self-intersection number is bounded above by L^2/4 + L/2 - 1, and that when L is even, this bound is sharp; in that case there are exactly four distinct classes attaining that bound. When L is odd, we establish a smaller, conjectured upper bound ((L^2 - 1)/4)) in certain cases; and there we show it is sharp. Furthermore, for the doubly-punctured plane, these self-intersection numbers are bounded below, by L/2 - 1 if L is even, (L - 1)/2 if L is odd; these bounds are sharp. }, added-at = {2010-11-01T12:28:52.000+0100}, author = {Chas, Moira and Phillips, Anthony}, biburl = {https://www.bibsonomy.org/bibtex/27bb63447de73536b72a8d9e4079c6d74/uludag}, description = {Self-intersection numbers of curves in the doubly-punctured plane}, interhash = {62e4a81725320b20775d9e76d1fa1ed3}, intrahash = {7bb63447de73536b72a8d9e4079c6d74}, keywords = {geodesic surfaces}, note = {cite arxiv:1001.4568 Comment: 15 pages, 8 figures}, timestamp = {2010-11-01T12:28:52.000+0100}, title = {Self-intersection numbers of curves in the doubly-punctured plane}, url = {http://arxiv.org/abs/1001.4568}, year = 2010 }