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.References
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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. - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 8391-8437
- MSC (2010): Primary 57M27, 57R58, 53D40, 57M25
- DOI: https://doi.org/10.1090/tran/8116
- MathSciNet review: 4177263