A Generalization of Turaev's Virtual String Cobracket
Patricia Cahn

TL;DR
This paper extends Turaev's virtual string cobracket with a new operation μ that provides improved bounds on the minimal self-intersection number of virtual strings, surpassing previous invariants in certain cases.
Contribution
It introduces a generalized operation μ for virtual strings that yields stronger bounds on self-intersections and utilizes Turaev's based matrices to characterize cases where μ is optimal.
Findings
μ provides a better bound than Turaev's cobracket for virtual strings.
μ's bound can be stronger than the invariant ρ from Turaev's based matrices.
An example demonstrates μ's bound surpasses previous bounds in specific cases.
Abstract
In a previous paper, we defined an operation 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. We show that 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 use Turaev's based matrices to describe strings such that gives a formula for the minimal self-intersection number . We also construct an example that shows the bound on the minimal self-intersection number given by is sometimes stronger than the bound given by Turaev's based matrix invariant.
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Taxonomy
TopicsGeometric and Algebraic Topology · Logic, programming, and type systems · semigroups and automata theory
