The Fukaya category of the pillowcase, traceless character varieties, and Khovanov cohomology

@article{Hedden2018TheFC,
  title={The Fukaya category of the pillowcase, traceless character varieties, and Khovanov cohomology},
  author={Matthew Hedden and Christopher Herald and Matthew Hogancamp and Paul A. Kirk},
  journal={arXiv: Geometric Topology},
  year={2018}
}
For a diagram of a 2-stranded tangle in the 3-ball we define a twisted complex of compact Lagrangians in the triangulated envelope of the Fukaya category of the smooth locus of the pillowcase. We show that this twisted complex is a functorial invariant of the isotopy class of the tangle, and that it provides a factorization of Bar-Natan's functor from the tangle cobordism category to chain complexes. In particular, the hom set of our invariant with a particular non-compact Lagrangian associated… 
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References

SHOWING 1-10 OF 37 REFERENCES
The pillowcase and perturbations of traceless representations of knot groups
We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli
Fukaya categories of the torus and Dehn surgery
TLDR
It is shown that A∞-structures on the graded algebra A formed by the cohomology of two basic objects in the Fukaya category of the punctured 2-torus are governed by just two parameters (m6, m8), extracted from the Hochschild cohmology of A.
The pillowcase and traceless representations of knot groups II: a Lagrangian-Floer theory in the pillowcase
We define an elementary relatively $\mathbb Z/4$ graded Lagrangian-Floer chain complex for restricted immersions of compact 1-manifolds into the pillowcase, and apply it to the intersection diagram
Khovanov's homology for tangles and cobordisms
We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots. By staying within a world of topological
Fukaya categories of symmetric products and bordered Heegaard-Floer homology
The main goal of this paper is to discuss a symplectic interpretation of Lipshitz, Ozsvath and Thurston's bordered Heegaard-Floer homology in terms of Fukaya categories of symmetric products and
A functor-valued invariant of tangles
We construct a family of rings. To a plane diagram of a tangle we associate a complex of bimodules over these rings. Chain homotopy equivalence class of this complex is an invariant of the tangle. On
Bordered Heegaard Floer homology
We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with
Khovanov homology from Floer cohomology
This paper realises the Khovanov homology of a link in S 3 S^3 as a Lagrangian Floer cohomology group, establishing a conjecture of Seidel and the second author. The starting point is
Fukaya categories and bordered Heegaard-Floer homology
We outline an interpretation of Heegaard-Floer homology of 3-manifolds (closed or with boundary) in terms of the symplectic topology of symmetric products of Riemann surfaces, as suggested by recent
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