2010/04/04 by Patricia Cahn, Cahn, Patricia · 1 citation
Mathematics · #17B62 (Secondary) #57M99 #57N05 (Primary) #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1004.0532
openalex publication_date 2010/04/04 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
Goldman and Turaev constructed a Lie bialgebra structure on the free\n\ℤ-module generated by free homotopy classes of loops on a surface.\nTuraev conjectured that his cobracket \Δ(\α) is zero if and only if\n\α is a power of a simple class. Chas constructed examples that show\nTuraev's conjecture is, unfortunately, false. We define an operation \μ in\nthe spirit of the Andersen-Mattes-Reshetikhin algebra of chord diagrams. The\nTuraev cobracket factors through \μ, so we can view \μ as a\ngeneralization of \Δ. We show that Turaev's conjecture holds when\n\Δ is replaced with \μ. We also show that \μ(\α) gives an\nexplicit formula for the minimum number of self-intersection points of a loop\nin \α. The operation \μ also satisfies identities similar to the\nco-Jacobi and coskew symmetry identities, so while \μ is not a cobracket,\n\μ behaves like a Lie cobracket for the Andersen-Mattes-Reshetikhin Poisson\nalgebra.\n