Configuration spaces of $\mathbb{C} \setminus k$

TL;DR

This paper reviews known formulas and methods for computing the standard and virtual Poincaré polynomials of configuration spaces of the plane with deleted points, comparing the results to deepen understanding.

Contribution

It compiles and compares existing formulas for Poincaré polynomials of configuration spaces of the punctured plane, clarifying their relationships.

Findings

Comparison of standard and virtual Poincaré polynomials

Clarification of formulas for configuration spaces of punctured plane

Insights into the structure of these polynomials

Abstract

In this note, we collect mostly known formulas and methods to compute the standard and virtual Poincar\'e polynomials of the configuration spaces of the plane $\mathbb{C} \setminus k$ with $k$ deleted points and compare the answers.