  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.
