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Transactions of the American Mathematical Society

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


Authors: Matthew Hedden, Christopher M. Herald, Matthew Hogancamp and Paul Kirk
Journal: Trans. Amer. Math. Soc. 373 (2020), 8391-8437
MSC (2010): Primary 57M27, 57R58, 53D40, 57M25
DOI: https://doi.org/10.1090/tran/8116
Published electronically: October 5, 2020
MathSciNet review: 4177263
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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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Additional Information

Matthew Hedden
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
MR Author ID: 769768
Email: mhedden@math.msu.edu

Christopher M. Herald
Affiliation: Department of Mathematics and Statistics, University of Nevada, Reno, Reno, Nevada 89557
MR Author ID: 357086
Email: herald@unr.edu

Matthew Hogancamp
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
MR Author ID: 948897
Email: m.hogancamp@northeastern.edu

Paul Kirk
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
MR Author ID: 266369
Email: pkirk@indiana.edu

Received by editor(s): October 9, 2018
Received by editor(s) in revised form: December 5, 2019
Published electronically: October 5, 2020
Additional Notes: The first author was partially supported by NSF CAREER grant DMS-1150872, NSF DMS-1709016, and an Alfred P. Sloan Research Fellowship during the course of this work.
The second and fourth authors were partially supported by Simons Collaboration Grants for Mathematicians.
The third author was partially supported by NSF DMS-1702274.
Article copyright: © Copyright 2020 American Mathematical Society