Almost simple linear graphs, homology cobordism and connected Heegaard Floer homology

TL;DR

This paper computes Heegaard Floer homologies for specific Brieskorn spheres using graph-based methods, revealing their role in the homology cobordism group and applications to knot concordance.

Contribution

It introduces effective computations of Floer homologies for almost simple linear graphs and demonstrates their significance in homology cobordism and knot theory.

Findings

Brieskorn spheres generate infinite rank summands in the homology cobordism group.

Effective computation methods for connected Heegaard Floer homologies.

Applications to classical knot and 2-knot concordance.

Abstract

Continuing our previous work, we effectively compute connected Heegaard Floer homologies of two families of Brieskorn spheres realized as the boundaries of almost simple linear graphs. Using Floer theoretic invariants of Dai, Hom, Stoffregen, and Truong, we show that these Brieskorn spheres also generate infinite rank summands in the homology cobordism group. Our computations also have applications to the concordance of classical knots and $0$-concordance of $2$-knots.