A link invariant from higher-dimensional Heegaard Floer homology

TL;DR

This paper introduces a higher-dimensional analogue of symplectic Khovanov homology, constructing a new link invariant using higher-dimensional Heegaard Floer homology applied to Milnor fibers of singularities.

Contribution

It defines a novel higher-dimensional link invariant based on symplectic geometry and Heegaard Floer homology, extending previous invariants to higher dimensions.

Findings

Proves invariance under arc slides and Markov stabilizations.

Establishes a new link invariant from higher-dimensional symplectic topology.

Connects Milnor fibers of singularities with link invariants.

Abstract

We define a higher-dimensional analogue of symplectic Khovanov homology. Consider the standard Lefschetz fibration $p\colon W\to D\subset\mathbb{C}$ of a $2n$-dimensional Milnor fiber of the $A_{2\kappa-1}$ singularity. We represent a link by a $\kappa$-strand braid, which is expressed as an element $h$ of the symplectic mapping class group $\mathrm{Symp}(W,\partial W)$. We then apply the higher-dimensional Heegaard Floer homology machinery to the pair $(\boldsymbol{a},h(\boldsymbol{a}))$, where $\boldsymbol{a}$ is a collection of $\kappa$ unstable manifolds of $W$ which are Lagrangian spheres. We prove its invariance under arc slides and Markov stabilizations, which shows that it is a link invariant. This work constitutes part of the author's PhD thesis.