
TL;DR
This paper generalizes Turaev's based matrix invariant from virtual 1-strings to virtual n-strings, providing a new algebraic tool for classifying virtual homotopy classes of multistring diagrams.
Contribution
It introduces a multistring based matrix construction that extends Turaev's invariant to virtual n-strings, solving an open problem in the field.
Findings
Constructed multistring based matrices for virtual n-strings.
Provided invariants distinguishing virtual homotopy classes.
Extended algebraic classification tools for virtual string theory.
Abstract
A virtual -string is a chord diagram with core circles and a collection of arrows between core circles. We consider virtual -strings up to virtual homotopy, compositions of flat virtual Reidemeister moves on chord diagrams. Given a virtual 1-string , Turaev associated a based matrix that encodes invariants of the virtual homotopy class of . We generalize Turaev's method to associate a multistring based matrix to virtual -strings, addressing an open problem of Turaev and constructing similar invariants for virtual homotopy classes of virtual -strings.
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