Knot Floer homology as immersed curves

To a nullhomologous knot $K$ in a 3-manifold $Y$, knot Floer homology associates a bigraded chain complex over $\mathbb{F}[U,V]$ as well as a collection of flip maps; we show that this data can be interpretted as a collection of decorated immersed curves in the marked torus. This is inspired by earlier work of the author with Rasmussen and Watson, showing that bordered Heegaard Floer invariants $\widehat{\mathit{CFD}}$ of manifolds with torus boundary can be interpreted in a similar way. Indeed, if we restrict the construction in this paper to the $UV = 0$ truncation of the knot Floer complex for knots in $S^3$ with $\mathbb{Z}/2\mathbb{Z}$ coefficients, which is equivalent to $\widehat{\mathit{CFD}}$ of the knot complement, we get precisely those curves; this paper then provides an entirely bordered-free treatment of those curves in the case of knot complements, which may appeal to readers unfamiliar with bordered Floer homology. On the other hand, the knot Floer complex is a stronger invariant than $\widehat{\mathit{CFD}}$ of the complement, capturing "minus" information while $\widehat{\mathit{CFD}}$ is only a "hat" flavor invariant. We show that this extra information is realized by adding an additional decoration, a bounding chain, to the immersed multicurves. We also give geometric surgery formulas, showing that $HF^-$ of rational surgeries on nullhomologous knots and the knot Floer complex of dual knots in integer surgeries can be computed by taking Floer homology of the appropriate decorated curves in the marked torus. A section of the paper is devoted to a giving a combinatorial construction of Floer homology of Lagrangians with bounding chains in marked surfaces, which may be of independent interest.