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A Generalization of the Turaev Cobracket for the Andersen-Mattes-Reshetikhin Algebra
by
Patricia Cahn
Dartmouth College
The Turaev cobracket Δ, defined on the free Z-module generated by free homotopy classes of loops on a surface, gives a lower bound on the number of self-intersection points of a loop α in a given free homotopy class. Turaev conjectured that Δ(α)=0 if and only if α is a power of a simple class. Chas constructed examples showing that this lower bound on the minimal self-intersection number is not an equality. These examples also disprove Turaev's conjecture. We construct a generalization μ of Δ, defined in the spirit of the Andersen-Mattes-Reshetikhin algebra of chord diagrams. We show that μ allows one to compute the minimal self-intersection number of a class α provided α is primitive (i.e., not a power of another class), and that an analogue of Turaev's conjecture holds for μ. In this talk, we will discuss the problem of computing the minimal self-intersection number of α when α is not primitive. We also extend μ to a subalgebra of the Andersen-Mattes-Reshetikhin algebra and explore its algebraic properties.
Paper reference: arXiv:1004.0532
Date received: April 16, 2010
Copyright © 2010 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbaf-08.