Complexity of virtual multistrings

TL;DR

This paper extends the understanding of virtual multistrings by analyzing their minimal representatives and the transformations connecting them, revealing new results and counterexamples in the topology of virtual strings.

Contribution

It generalizes Cahn's genus and crossing minimization results to connected non-parallel n-strings and clarifies the conditions under which Kadokami's claim holds or fails.

Findings

Virtual n-strings can be minimized in genus and crossings similarly to 1-strings.

Kadokami's claim holds for connected non-parallel n-strings.

Counterexamples exist for 3-strings, showing the claim does not hold universally.

Abstract

A virtual $n$-string $\alpha$ is a collection of $n$ oriented smooth generic loops on a surface $M$. A stabilization of $\alpha$ is a surgery that results in attaching a handle to $M$ along disks avoiding $\alpha$, and the inverse operation is a destabilization of $\alpha$. We consider virtual $n$-strings up to virtual homotopy, i.e., sequences of stabilizations, destabilizations, and homotopies of $\alpha$. Recently, Cahn proved that any virtual $1$-string can be virtually homotoped to a genus-minimal and crossing-minimal representative by monotonically decreasing both genus and the number of self-intersections. We generalize her result to the case of connected non-parallel $n$-strings. Cahn also proved that any two crossing-irreducible representatives of a virtual $1$-string are related by Type 3 moves, stabilizations, and destabilizations. Kadokami claimed that this held for virtual $n$-strings in general, but Gibson found a counterexample for $5$-strings. We show that Kadokami's statement holds for connected non-parallel $n$-strings and exhibit a counterexample for $3$-strings.