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A Generalization of Turaev's Cobracket and the Minimal Self-Intersection Number of a Virtual String
by
Patricia Cahn
Dartmouth College
Turaev introduced a Lie cobracket on the free Z-module generated by nontrivial free homotopy classes of loops on a surface. Turaev's cobracket gives a lower bound on the minimum number of self-intersection points of a loop in a given free homotopy class. We introduce an operation μ which can be viewed as a generalization of Turaev's cobracket. We show that this operation gives an exact formula for the minimal number of self-intersection points of a loop in a given free homotopy class. Both Turaev's cobracket and μ can be extended to virtual strings, and both operations give a lower bound on the number of self-intersection points of a virtual string in a given virtual homotopy class. We show that the bound given by μ is similar to a bound given by Turaev's based matrix invariant, and is stronger than the bound given by Turaev's cobracket. We also show that μ gives an explicit formula for the minimal number of self-intersection points of a virtual string in certain virtual homotopy classes.
Paper reference: arXiv:1004.0532
Date received: November 17, 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 # cbbr-15.