For the null homologous knot $K$ of the 3-manifold $Y$ , the knot Floer homology relates a weighted chain complex on $\mathbb{F}[U,V]A collection of $ and flip maps; we show that this data can be interpreted as a set of decorated immersion curves within the marked torus. This is inspired by the authors’ previous work with Rasmussen and Watson, stating that the bounded Heegaard Floer invariant $\widehat{\mathit{CFD}}$ on manifolds with torus boundaries can be interpreted in a similar way. is shown. In fact, if we limit the construction of this paper to the $UV = 0$ truncation of the not-Flore complex of the knots of $S^3$ with $\mathbb{Z}/2\mathbb{Z}$ coefficients, then will be The $\widehat{\mathit{CFD}}$ equivalent of knot’s complement gives us exactly these curves. This paper then provides a completely boundless treatment of these curves in the case of knot complements. This may appeal to readers unfamiliar with his Floer homology with boundaries. On the other hand, the not-Floer complex is a stronger invariant than the complement $\widehat{\mathit{CFD}}$, where $\widehat{\mathit{CFD}}$ is just “, whereas “minus It captures the information of the “. Hat” flavor is immutable. We show that this additional information is achieved by adding an additional adornment, the bounding chain, to the embedded multicurves. We also give geometric operation formulas, where $HF^-$ for rational operations on null homologous knots and knot-Floer complexes for dual knots in integer operations give Floer homology of appropriate ornamental curves in the marked torus. Indicates that it can be calculated by taking A section of this paper is devoted to giving the combinatorial construction of his Floer homology of Lagrangians with boundary chains on the marked surfaces, which could be of independent interest.

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