Minimizing intersection points of curves under virtual homotopy

TL;DR

This paper establishes a method to determine the minimal intersection points of curves on surfaces under virtual homotopy using algebraic invariants, generalizing known brackets and cobrackets.

Contribution

It introduces generalized Poisson brackets and cobrackets to compute minimal intersections and self-intersections in virtual homotopy classes of curves.

Findings

Minimal intersection points correspond to the number of terms in a generalized Poisson bracket.

Minimal self-intersections are counted by a generalized Cahn cobracket.

Provides algebraic tools for analyzing virtual homotopy classes of curves.

Abstract

A flat virtual link is a finite collection of oriented closed curves $\mathfrak L$ on an oriented surface $M$ considered up to virtual homotopy, i.e., a composition of elementary stabilizations, destabilizations, and homotopies. Specializing to a pair of curves $(L_1,L_2)$, we show that the minimal number of intersection points of curves in the virtual homotopy class of $(L_1, L_2)$ equals to the number of terms of a generalization of the Anderson--Mattes--Reshetikhin Poisson bracket. Furthermore, considering a single curve, we show that the minimal number of self-intersections of a curve in its virtual homotopy class can be counted by a generalization of the Cahn cobracket.