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A generalization of Turaev's virtual string cobracket and self-intersections of virtual strings

Abstract

Previously we defined an operation µ that generalizes Turaev’s cobracket for loops on a surface. We showed that, in contrast to the cobracket, this operation gives a formula for the minimum number of self-intersections of a loop in a given free homotopy class. In this paper we consider the corresponding question for virtual strings, and conjecture that µ gives a formula for the minimum number of self-intersection points of a virtual string in a given virtual homotopy class. To support the conjecture, we show that µ gives a bound on the minimal self-intersection number of a virtual string which is stronger than a bound given by Turaev’s virtual string cobracket. We also use Turaev’s based matrices to describe a large set of strings α such that µ gives a formula for the minimal self-intersection number α. Finally, we construct an example that shows the bound on the minimal self-intersection number given by µ is always at least as good as, and sometimes stronger than, the bound ρ given by Turaev’s based matrix invariant

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