Hilbert polynomials of configuration spaces over graphs of circumference at most 1

TL;DR

This paper studies the Betti numbers of configuration spaces over certain graphs, showing they are polynomial in the number of points and providing explicit formulas and combinatorial descriptions for these polynomials.

Contribution

It introduces the Hilbert polynomial for configuration spaces over graphs with circumference at most 1 and derives explicit formulas based on graph decompositions.

Findings

Betti numbers are polynomial in the number of points

Explicit formulas for Hilbert polynomials are provided

Coefficients have a combinatorial description

Abstract

The $ k $-configuration space $ B_k\Gamma $ of a topological space $ \Gamma $ is the space of sets of $ k $ distinct points in $ \Gamma $. In this paper, we consider the case where $ \Gamma $ is a graph of circumference at most $1$. We show that for all $ k\ge0 $, the $ i $-th Betti number of $ B_k\Gamma $ is given by a polynomial $P_\Gamma^i(k)$ in $ k $, called the Hilbert polynomial of $ \Gamma $. We find an expression for the Hilbert polynomial $P_\Gamma^i(k)$ in terms of those coming from the canonical $1$-bridge decomposition of $ \Gamma $. We also give a combinatorial description of the coefficients of $P_\Gamma^i(k)$.