A link invariant from higher-dimensional Heegaard Floer homology

Abstract

We define a higher-dimensional analogue of symplectic Khovanov homology. Consider the standard Lefschetz fibration $p:W\to D\subset \mathbb{C}$ 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 $(\mathbf{a},h(\mathbf{a}))$, where $\mathbf{a}$ 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.