Coverings, composites and cables of virtual strings

TL;DR

This paper explores operations on virtual strings—covering, composition, and cabling—and examines their effects on invariants, including characterizing fixed points and correcting previous results about based matrices.

Contribution

It introduces and analyzes three operations on virtual strings, clarifies the relationship of based matrices in composites, and studies invariant behavior under cabling.

Findings

Virtual strings unchanged by covering operation identified.

Relationship between based matrices of composite strings clarified.

Behavior of Turaev invariants under cabling studied.

Abstract

A virtual string can be defined as an equivalence class of planar diagrams under certain kinds of diagrammatic moves. Virtual strings are related to virtual knots in that a simple operation on a virtual knot diagram produces a diagram for a virtual string. In this paper we consider three operations on a virtual string or virtual strings which produce another virtual string, namely covering, composition and cabling. In particular we study virtual strings unchanged by the covering operation. We also show how the based matrix of a composite virtual string is related to the based matrices of its components, correcting a result by Turaev. Finally we investigate what happens under cabling to some invariants defined by Turaev.