# Models for Configuration Space in a Simplicial Complex

## TL;DR

This paper develops combinatorial models for the configuration space of a simplicial complex, including local and global models, and applies these to study braid groups associated with a nodal curve.

## Contribution

It introduces novel combinatorial models for configuration spaces in simplicial complexes and their local variants, and applies these models to analyze braid groups of a specific singular algebraic curve.

## Key findings

- Presented a combinatorial model for local configuration space using posets.
- Developed a global configuration space model based on a deleted subcomplex.
- Derived presentations for braid groups associated with the nodal curve.

## Abstract

We produce combinatorial models for configuration space in a simplicial complex, and for configurations near a single point ("local configuration space.") The model for local configuration space is built out of the poset of poset structures on a finite set. The model for global configuration space relies on a combinatorial model for a simplicial complex with a deleted subcomplex. By way of application, we study the nodal curve $y^2 z = x^3 + x^2 z$, obtaining a presentation for its two-strand braid group, a conjectural presentation for its three-strand braid group, and presentations for its two- and three-strand local braid groups near the singular point.