Explicit Betti Numbers for Skeletons of Chordal Clique Complexes and Their Alexander Duals

TL;DR

This paper provides explicit formulas for Betti numbers, regularity, and other invariants of skeletons of chordal clique complexes and their Alexander duals, advancing understanding of their algebraic and topological properties.

Contribution

It introduces explicit graded Betti numbers and formulas for invariants of these complexes and their duals, linking algebraic, topological, and combinatorial aspects.

Findings

Explicit Betti numbers for all skeletons

Regularity is k+1 for k-skeletons

Derived combinatorial binomial identities

Abstract

We study the homological properties of $\Delta_{\mathbf{r}}(n_1, \dots, n_e)$, a simplicial complex formed by sequentially gluing complete graphs along $(r_i-1)$-simplices. This construction generates precisely the chordal clique complexes, whose Stanley-Reisner ideals admit 2-linear resolutions. By computing the $f$-vector and evaluating the Hilbert series, we establish explicit graded Betti numbers for all $k$-skeletons. We show that the regularity of these skeletons is $k+1$ and the projective dimension stabilizes at $N_{\mathbf{r}} - r_{\min} - 1$ for $k \ge r_{\min}$, providing a complete classification of when the complex is Cohen-Macaulay, sequentially Cohen-Macaulay, or initially Cohen-Macaulay. We also obtain explicit formulas for the ring multiplicity and reduced Euler characteristic. Applying Alexander duality, we derive the $f$-vector, rational $h$-polynomial, and exact graded Betti numbers of the dual and its skeletons. Furthermore, analyzing these dual skeletons yields a family of complexes that resolve recent open bounds on regularity. Finally, equating the topological and rational evaluations of the Hilbert series produces a new family of combinatorial binomial identities.