A Generalization of the Turaev Cobracket and the Minimal Self-Intersection Number of a Curve on a Surface

Patricia Cahn

Knots in Washington XXIX; 30 years of quandles, 10 years of Khovanov homology
December 4-6, 2009
George Washington University
Washington, DC, USA

Organizers
Jozef H. Przytycki (GWU), Yongwu Rong (GWU, NSF), Radmila Sazdanovic (GWU), Alexander Shumakovitch (GWU), Hao Wu (GWU)

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A Generalization of the Turaev Cobracket and the Minimal Self-Intersection Number of a Curve on a Surface
by
Patricia Cahn
Dartmouth College

Goldman and Turaev constructed a Lie bialgebra structure on the vector space generated by free homotopy classes of loops on a surface. The Turaev cobracket gives a lower bound on the minimal number of self-intersection points of a loop in a given homotopy class. Chas proved that this lower bound is not sharp by providing examples of homotopy classes with zero cobracket that do not contain a simple representative. The Turaev cobracket factors through another operation μ, defined in the spirit of the Andersen-Mattes-Reshetikhin algebra. The operation μ also gives a lower bound on the minimal number of self-intersection points of a loop in a given homotopy class. We will show that this lower bound is sharp for homotopy classes α that do not contain a power of a curve. In particular, we show that the sum of the absolute values of the coefficients of μ(α) is always twice the minimal number of self-intersection points of a loop in α.

Date received: October 28, 2009