Patterns in the Lattice Homology of Seifert Homology Spheres

Karthik Seetharaman; William Yue; Isaac Zhu

paper

Patterns in the Lattice Homology of Seifert Homology Spheres

arXiv:2110.13405

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 Ozsváth-Szabó and Némethi. We reprove the fact that the $d$ -invariants of Seifert homology spheres $Σ (a_{1}, a_{2}, \dots, a_{n})$ and $Σ (a_{1}, a_{2}, \dots, a_{n} + a_{1} a_{2} \dots a_{n - 1})$ are the same using an explicit understanding of the behavior of the numerical semigroup minimally generated by $a_{1} a_{2} \dots a_{n} / a_{i}$ 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 $Σ (a_{1}, a_{2}, \dots, a_{n})$ and $Σ (a_{1}, a_{2}, \dots, a_{n} + 2 a_{1} a_{2} \dots a_{n - 1})$ are the same.

25 pages, 8 figures

References in corpus (1)

Heegaard-Floer Homology and a family of Brieskorn spheres