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arXiv · 2309.13241

A link invariant from higher-dimensional Heegaard Floer homology

Tianyu Yuan
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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.

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Tianyu Yuan. 2023-09-23. A link invariant from higher-dimensional Heegaard Floer homology. https://arxiv.org/abs/2309.13241

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