Knot Floer homology

Definition

Let $K \subset S^3$ be an oriented tame knot. Let $(\Sigma, \boldsymbol{\alpha}, \boldsymbol{\beta}, w, z)$ be a doubly pointed Heegaard diagram for $(S^3, K)$, built from a Heegaard splitting of genus $g$. Here $\boldsymbol{\alpha} = (\alpha_1, \ldots, \alpha_g)$ and $\boldsymbol{\beta} = (\beta_1, \ldots, \beta_g)$ are attaching curves, and $w, z$ are basepoints off them. Let $\mathbb{T}_\alpha = \alpha_1 \times \cdots \times \alpha_g$ and $\mathbb{T}_\beta = \beta_1 \times \cdots \times \beta_g$, viewed as tori in the symmetric product $\operatorname{Sym}^g(\Sigma)$. Let $\widehat{CFK}(K)$ be the $\F_2$-vector space with basis $\mathbb{T}_\alpha \cap \mathbb{T}_\beta$. Its differential is $$\partial x = \sum_{y} \# \big( \mathcal{M}(\phi) / \R \big) \, y,$$ summed over $y$ and over Whitney disk classes $\phi$ from $x$ to $y$ with Maslov index $1$ and $n_w(\phi) = n_z(\phi) = 0$. Here $\mathcal{M}(\phi)$ is the moduli space of holomorphic representatives of $\phi$, counted modulo $2$. The knot Floer homology $\widehat{HFK}(K)$ is the homology of $(\widehat{CFK}(K), \partial)$.