Curve diagram for Artin group of type B

TL;DR

This paper introduces a geometric approach using curve diagrams to analyze Artin groups of type B, defining new labelings that detect Garside lengths without relying on traditional algebraic methods.

Contribution

It develops a novel geometric framework with winding number and wall crossing labelings to determine Garside lengths in Artin groups of type B, avoiding Garside theory machinery.

Findings

Labelings detect classical and dual Garside lengths

Method is purely geometric, not relying on normal forms

Provides new insights into Artin group structure

Abstract

We develop a theory of curve diagrams for Artin groups of type B. We define the winding number labeling and the wall crossing labeling of curve diagrams, and show that these labelings detect the classical and the dual Garside length, respectively. A remarkable point is that our argument does not require Garside theory machinery like normal forms, and is more geometric in nature.