 

Patricia Cahn
A generalization of the Turaev cobracket and the minimal selfintersection number of a curve on a surface view print


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


Abstract
Goldman and Turaev constructed a Lie bialgebra structure on the free Zmodule 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 AndersenMattesReshetikhin 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 selfintersection points of a loop in α. The operation μ also satisfies identities similar to the
coJacobi and coskew symmetry identities, so while μ is not a cobracket, μ behaves like a Lie cobracket for the AndersenMattesReshetikhin Poisson algebra.


Author information
Patricia Cahn, Department of Mathematics, University of Pennsylvania, David Rittenhouse Lab. 209 South 33rd Street, Philadelphia, PA 191046395, USA
pcahn@math.upenn.edu

