Self-Intersection Numbers of Curves on the Punctured Torus

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Self-Intersection Numbers of Curves on the Punctured Torus
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 counting, for each non-negative integer, how many length-L classes have that self...
Moira Chas, Anthony Phillips
Added 10 Dec 2010
Updated 10 Dec 2010
Type Journal
Year 2010
Where EM
Authors Moira Chas, Anthony Phillips
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