Curve Diagrams, Laminations, and the Geometric Complexity of Braids

TL;DR

This paper investigates the geometric complexity of braids through curve diagrams and introduces a geometric generating function, revealing its non-rational nature for three-strand braids despite the standard complexity being easier to compute.

Contribution

It defines the geometric generating function for braid groups and explicitly computes it for three strands, showing it is neither rational nor algebraic.

Findings

The geometric generating function for three-strand braids is neither rational nor algebraic.

Standard braid complexity is harder to compute than geometric complexity.

The geometric generating function is not holonomic.

Abstract

Braids can be represented geometrically as curve diagrams. The geometric complexity of a braid is the minimal complexity of a curve diagram representing it. We introduce and study the corresponding notion of geometric generating function. We compute explicitly the geometric generating function for the group of braids on three strands and prove that it is neither rational nor algebraic, nor even holonomic. This result may appear as counterintuitive. Indeed, the standard complexity (due to the Artin presentation of braid groups) is algorithmically harder to compute than the geometric complexity, yet the associated generating function for the group of braids on three strands is rational.