A generalization of the Turaev cobracket and the minimal self-intersection number of a curve on a surface

作者：Cahn Patricia^*

来源：The New York Journal of Mathematics, 2013, 19: 253-283.

• Self-intersections
• curves on surfaces
• free homotopy classes
• Lie bialgebras

Goldman and Turaev constructed a Lie bialgebra structure on the free Z-module generated by free homotopy classes of loops on a surface. Turaev conjectured that his cobracket Delta(alpha) is zero if and only if alpha is a power of a simple class. Chas constructed examples that show Turaev's conjecture is, unfortunately, false. We define an operation mu in the spirit of the Andersen-Mattes-Reshetikhin algebra of chord diagrams. The Turaev cobracket factors through mu, so we can view as a generalization of Delta. We show that Turaev's conjecture holds when Delta is replaced with mu. We also show that mu(alpha) gives an explicit formula for the minimum number of self-intersection points of a loop in alpha. The operation mu also satisfies identities similar to the co-Jacobi and coskew symmetry identities, so while mu is not a cobracket, mu behaves like a Lie cobracket for the Andersen-Mattes-Reshetikhin Poisson algebra.

• 出版日期2013