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  "body": "\n## Definition\n\nLet $K \\subset S^3$ be an oriented tame [[knot|QYEAZTYGPT]].\nLet $(\\Sigma, \\boldsymbol{\\alpha}, \\boldsymbol{\\beta}, w, z)$ be a doubly pointed Heegaard diagram for $(S^3, K)$, built from a [[Heegaard splitting|EDE9919B5R]] of genus $g$.\nHere $\\boldsymbol{\\alpha} = (\\alpha_1, \\ldots, \\alpha_g)$ and $\\boldsymbol{\\beta} = (\\beta_1, \\ldots, \\beta_g)$ are attaching curves, and $w, z$ are basepoints off them.\nLet $\\mathbb{T}_\\alpha = \\alpha_1 \\times \\cdots \\times \\alpha_g$ and $\\mathbb{T}_\\beta = \\beta_1 \\times \\cdots \\times \\beta_g$, viewed as tori in the symmetric product $\\operatorname{Sym}^g(\\Sigma)$.\nLet $\\widehat{CFK}(K)$ be the $\\F_2$-vector space with basis $\\mathbb{T}_\\alpha \\cap \\mathbb{T}_\\beta$.\nIts differential is\n$$\\partial x = \\sum_{y} \\# \\big( \\mathcal{M}(\\phi) / \\R \\big) \\, y,$$\nsummed over $y$ and over Whitney disk classes $\\phi$ from $x$ to $y$ with Maslov index $1$ and $n_w(\\phi) = n_z(\\phi) = 0$.\nHere $\\mathcal{M}(\\phi)$ is the moduli space of holomorphic representatives of $\\phi$, counted modulo $2$.\nThe *knot Floer homology* $\\widehat{HFK}(K)$ is the homology of $(\\widehat{CFK}(K), \\partial)$.\n\n## Remarks\n\nIt carries a Maslov grading $m$ and an Alexander grading $a$, and up to isomorphism it does not depend on the diagram.\nThe graded Euler characteristic is the Conway-normalized [[Alexander polynomial|JKA25KDPQP]]: $\\sum_{m,a} (-1)^m \\dim \\widehat{HFK}_m(K, a) \\, t^a = \\Delta_K(t)$.\nExample: the [[unknot|1TQ06R4Q66]] has a genus-$0$ diagram on $S^2$ with one generator, so $\\widehat{HFK}$ is $\\F_2$ in bidegree $(0, 0)$.\nNon-example: if the condition $n_z(\\phi) = 0$ is dropped, the complex computes $\\widehat{HF}(S^3) \\cong \\F_2$, which forgets the knot.\nOzsváth and Szabó proved that the Seifert genus of $K$ is the largest $a$ with $\\widehat{HFK}(K, a) \\ne 0$.\n",
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