Patterns in the Lattice Homology of Seifert Homology Spheres

TL;DR

This paper investigates lattice homology invariants of Seifert homology spheres, demonstrating invariance of certain $d$-invariants and maximal monotone subroots under specific modifications, thereby deepening understanding of their homology cobordism properties.

Contribution

It provides explicit proofs of invariance of $d$-invariants and maximal monotone subroots for Seifert homology spheres under particular parameter changes, using combinatorial and algebraic techniques.

Findings

$d$-invariants are invariant under specific sphere modifications

Maximal monotone subroots remain unchanged under certain parameter shifts

Lattice homology invariants reveal structural stability of Seifert spheres

Abstract

In this paper, we study various homology cobordism invariants for Seifert fibered integral homology 3-spheres derived from Heegaard Floer homology. Our main tool is lattice homology, a combinatorial invariant defined by Ozsv\'ath-Szab\'o and N\'emethi. We reprove the fact that the $d$-invariants of Seifert homology spheres $\Sigma(a_1,a_2,\dots,a_n)$ and $\Sigma(a_1,a_2,\dots,a_n+a_1a_2\cdots a_{n-1})$ are the same using an explicit understanding of the behavior of the numerical semigroup minimally generated by $a_1a_2\cdots a_n/a_i$ for $i\in[1,n]$. We also study the maximal monotone subroots of the lattice homologies, another homology cobordism invariant introduced by Dai and Manolescu. We show that the maximal monotone subroots of the lattice homologies of Seifert homology spheres $\Sigma(a_1,a_2,\dots,a_n)$ and $\Sigma(a_1,a_2,\dots,a_n+2a_1a_2\cdots a_{n-1})$ are the same.