Associated to the diagrams I and II of the figure there are operators c n : H ⊗n → H and s n : H → H ⊗n , n = 2, 3, 4, .. These operators satisfy various relations e.g. c 2 and s 2 satisfy the relations-Jacobi, coJacobi, and Drinfeld compatibility which utilize diagrams III, IV, V, and VI.
Goldman [7] and Turaev [11] found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of curves on a surface. When the surface has non-empty boundary, this vector space has a basis of cyclic reduced words in the generators of the fundamental group and their inverses. We give a combinatorial algorithm to compute this Lie bialgebra on this vector space of cyclic words. Using this presentation, we prove a variant of Goldman's result relating the bracket to disjointness of curve representatives when one of the classes is simple. We exhibit some examples we found by programming the algorithm which answer negatively Turaev's question about the characterization of simple curves in terms of the cobracket. Further computations suggest an alternative characterization of simple curves in terms of the bracket of a curve and its inverse. Turaev's question is still open in genus zero.2000 Mathematics Subject Classification: Primary 57M99, Secondary 17B62.
This paper is a consequence of the close connection between combinatorial group theory and the topology of surfaces.In the eighties Goldman discovered a Lie algebra structure on the vector space generated by the free homotopy classes of oriented curves on an oriented surface. The Lie bracket [a,b] is defined as the signed sum over the intersection points of a and b of the loop product of at the intersection points.If one of the classes has a simple representative we give a combinatorial group theory description of the terms of the Lie bracket and prove that this bracket has as many terms, counted with multiplicity, as the minimal number of intersection points of a and b. In other words the bracket with a simple element has no cancellation and determines minimal intersection numbers. We show that analogous results hold for the Lie bracket (also discovered by Goldman) of unoriented curves. We give three applications: a factorization of Thurston's map defining the boundary of Teichmüller space, various decompositions of the underlying vector space of conjugacy classes into ad invariant subspaces and a connection between bijections of the set of conjugacy classes of curves on a surface preserving the Goldman bracket and the mapping class group.
Oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses. By self-intersection number one means the minimum number of transversal self-intersection points of representatives of the class. We prove that if a class is chosen at random from among all classes of m letters, then for large m the distribution of the self-intersection number approaches the Gaussian distribution. The theorem was strongly suggested by a computer experiment with four million curves producing a very nearly Gaussian distribution.
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