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Geometric interpretation of the differential in A bar r A . 

Geometric interpretation of the differential in A bar r A . 

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We construct an algebraic version of Lagrangian Floer homology for immersed curves inside the pillowcase. We first associate to the pillowcase an algebra A. Then to an immersed curve L inside the pillowcase we associate an A infinity module M(L) over A. Then we prove that Lagrangian Floer homology HF(L,L') is isomorphic to a suitable algebraic pair...

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... (See [33,57,64] for analogous results for closed 3-manifolds.) In a different direction, Artem Kotelskiy, Watson, and Zibrowius showed that, for 4-ended tangles, Bar-Natan's extension of Khovanov homology to tangles can also be interpreted as an immersed curve in a 4-punctured sphere [83], and this immersed curve in fact agrees [84] with an invariant introduced by Hedden, Christopher Herald, Matthew Hogancamp, and Paul Kirk, inspired by instanton link homology [62] (see also [63,82]). ...
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Bordered Floer homology is an invariant for 3-manifolds with boundary, defined by the authors in 2008. It extends the Heegaard Floer homology of closed 3-manifolds, defined in earlier work of Zolt\'an Szab\'o and the second author. In addition to its conceptual interest, bordered Floer homology also provides powerful computational tools. This survey outlines the theory, focusing on recent developments and applications.
... In words, the Khovanov homology of the link arising as the union of a tangle T with the mirror of a tangle V can be recovered, up to tensoring with a particular two dimensional vector space, as the cohomology of the space of morphisms between the twisted complexes we associate to V and T , respectively. We should point out that similar results in these directions, and more precise comparisons with bordered Floer homology, have been obtained independently by Kotelskiy [21,20]. ...
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