Multistring based matrices

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 $n$-string is a chord diagram with $n$ core circles and a collection of arrows between core circles. We consider virtual $n$-strings up to virtual homotopy, compositions of flat virtual Reidemeister moves on chord diagrams. Given a virtual 1-string $\alpha$, Turaev associated a based matrix that encodes invariants of the virtual homotopy class of $\alpha$. We generalize Turaev's method to associate a multistring based matrix to virtual $n$-strings, addressing an open problem of Turaev and constructing similar invariants for virtual homotopy classes of virtual $n$-strings.