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

Hedden, Matthew; Herald, Christopher; Hogancamp, Matthew; Kirk, Paul

doi:10.1090/tran/8116

The Fukaya category of the pillowcase, traceless character varieties, and Khovanov cohomology
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by Matthew Hedden, Christopher M. Herald, Matthew Hogancamp and Paul Kirk PDF

Trans. Amer. Math. Soc. 373 (2020), 8391-8437 Request permission

Abstract:

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 to the trivial tangle is naturally isomorphic to the reduced Khovanov chain complex of the closure of the tangle. Our construction comes from the geometry of traceless $SU(2)$ character varieties associated to resolutions of the tangle diagram, and was inspired by Kronheimer and Mrowka’s singular instanton link homology.

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