New York Journal of Mathematics
Volume 19 (2013) 253-283


Patricia Cahn

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

view    print

Published: June 5, 2013
Keywords: Self-intersections, curves on surfaces, free homotopy classes, Lie bialgebras
Subject: 57N05, 57M99 (primary), 17B62 (secondary)

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 Δ(α) is zero if and only if α is a power of a simple class. Chas constructed examples that show Turaev's conjecture is, unfortunately, false. We define an operation μ in the spirit of the Andersen-Mattes-Reshetikhin algebra of chord diagrams. The Turaev cobracket factors through μ, so we can view μ as a generalization of Δ. We show that Turaev's conjecture holds when Δ is replaced with μ. We also show that μ(α) gives an explicit formula for the minimum number of self-intersection points of a loop in α. The operation μ also satisfies identities similar to the co-Jacobi and coskew symmetry identities, so while μ is not a cobracket, μ behaves like a Lie cobracket for the Andersen-Mattes-Reshetikhin Poisson algebra.

Author information

Patricia Cahn, Department of Mathematics, University of Pennsylvania, David Rittenhouse Lab. 209 South 33rd Street, Philadelphia, PA 19104-6395, USA