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GENERATORS, RELATIONS, AND HOMOLOGY FOR OZSVÁTH–SZABÓ’S KAUFFMAN-STATES ALGEBRAS

Part of: Homological methods Low-dimensional topology Differential topology

Published online by Cambridge University Press:  17 April 2020

ANDREW MANION Open the ORCID record for ANDREW MANION [Opens in a new window] ,
MARCO MARENGON  and
MICHAEL WILLIS
ANDREW MANION
Affiliation:
Department of Mathematics, USC, 3620 S. Vermont Ave., Los Angeles, CA 90089, USA email [email protected]
MARCO MARENGON
Affiliation:
Department of Mathematics, UCLA, 520 Portola Plaza, Los Angeles, CA 90095, USA email [email protected]
MICHAEL WILLIS
Affiliation:
Department of Mathematics, UCLA, 520 Portola Plaza, Los Angeles, CA 90095, USA email [email protected]
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Abstract

We give a generators-and-relations description of differential graded algebras recently introduced by Ozsváth and Szabó for the computation of knot Floer homology. We also compute the homology of these algebras and determine when they are formal.

MSC classification

Primary: 57R58: Floer homology
Secondary: 16E45: Differential graded algebras and applications 57M27: Invariants of knots and 3-manifolds 57R56: Topological quantum field theories
Type
Article
Information
Nagoya Mathematical Journal , Volume 244 , December 2021 , pp. 60 - 118
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Copyright
© 2020 Foundation Nagoya Mathematical Journal

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Footnotes

A.M. was supported by an NSF MSPRF fellowship, grant number DMS-1502686. M.M. was supported by the NSF FRG grant DMS-1563615. M.W. was supported by the NSF FRG grant DMS-1563615.

References

Alishahi, A. and Dowlin, N., A link invariant related to Khovanov homology and knot Floer homology, Preprint, 2018. arXiv:1810.13406.Google Scholar
Bar-Natan, D., Fast Khovanov homology computations , J. Knot Theory Ramifications 16(3) (2007), 243–255.CrossRefGoogle Scholar
Dunfield, N. M., Gukov, S. and Rasmussen, J., The superpolynomial for knot homologies , Experim. Math. 15(2) (2006), 129–159.CrossRefGoogle Scholar
Drinfeld, V., DG quotients of DG categories , J. Algebra 272(2) (2004), 643–691.10.1016/j.jalgebra.2003.05.001CrossRefGoogle Scholar
Eilenberg, S. and Kelly., G. M., “ Closed categories ”, in Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), Springer, New York, 1966, 421–562.CrossRefGoogle Scholar
Freedman, M., Gompf, R., Morrison, S. and Walker, K., Man and machine thinking about the smooth 4-dimensional Poincaré conjecture , Quantum Topol. 1(2) (2010), 171–208.CrossRefGoogle Scholar
Hanselman, J., Rasmussen, J. and Watson, L., Bordered Floer homology for manifolds with torus boundary via immersed curves, Preprint, 2016, arXiv:1604.03466.Google Scholar
Kauffman, L. H., Formal Knot Theory, Mathematical Notes 30 , Princeton University Press, Princeton, NJ, 1983.Google Scholar
Khovanov, M., A categorification of the Jones polynomial , Duke Math. J. 101(3) (2000), 359–426.CrossRefGoogle Scholar
Lefevre-Hasegawa, K., Sur les A-infini catégories, PhD Thesis, Université Paris-Diderot – Paris VII, 2003.Google Scholar
Lipshitz, R., Ozsváth, P. S. and Thurston, D. P., A bordered $HF^{-}$ algebra for the torus, in preparation.Google Scholar
Lipshitz, R., Ozsváth, P. S. and Thurston, D. P., Bimodules in bordered Heegaard Floer homology , Geom. Topol. 19(2) (2015), 525–724.CrossRefGoogle Scholar
Lipshitz, R., Ozsváth, P. S. and Thurston, D. P., Bordered Heegaard Floer homology , Mem. Amer. Math. Soc. 254(1216) (2018), viii+279.Google Scholar
Lunts, V. A., Formality of DG algebras (after Kaledin) , J. Algebra 323(4) (2010), 878–898.CrossRefGoogle Scholar
Manion, A., Khovanov–Seidel quiver algebras and Ozsváth–Szabó’s bordered theory , J. Algebra 488 (2017), 110–144.CrossRefGoogle Scholar
Manion, A., On the decategorification of Ozsváth and Szabó’s bordered theory for knot Floer homology , Quantum Topol. 10(1) (2019), 77–206.CrossRefGoogle Scholar
Manion, A., Marengon, M. and Willis, M., Bimodules over strands algebras and Ozsváth–Szabó’s Kauffman states functor, in preparation.Google Scholar
Manion, A., Marengon, M. and Willis, M., Strands algebras and Ozsváth–Szabó’s Kauffman states functor, Algebr Geom. Topol., Preprint, 2019. arXiv:1903.05655.Google Scholar
Manion, A. and Rouquier, R., Higher representations and arc algebras, in preparation.Google Scholar
Ozsváth, P. S. and Szabó, Z., Knot Floer homology calculator, http://www.math.princeton.edu/ szabo/HFKcalc.html.Google Scholar
Ozsváth, P. S. and Szabó, Z., The Pong algebra, in preparation.Google Scholar
Ozsváth, P. S. and Szabó, Z., Holomorphic disks and knot invariants , Adv. Math. 186(1) (2004), 58–116.10.1016/j.aim.2003.05.001CrossRefGoogle Scholar
Ozsváth, P. S. and Szabó, Z., Holomorphic disks and three-manifold invariants: properties and applications , Ann. of Math. (2) 159(3) (2004), 1159–1245.Google Scholar
Ozsváth, P. S. and Szabó, Z., Holomorphic disks and topological invariants for closed three-manifolds , Ann. of Math. (2) 159(3) (2004), 1027–1158.CrossRefGoogle Scholar
Ozsváth, P. S. and Szabó, Z., Kauffman states, bordered algebras, and a bigraded knot invariant , Adv. Math. 328 (2018), 1088–1198; https://arxiv.org/abs/1603.06559.CrossRefGoogle Scholar
Ozsváth, P. S. and Szabó, Z., Algebras with matchings and knot Floer homology, Preprint, 2019. arXiv:1912.01657.CrossRefGoogle Scholar
Ozsváth, P. S. and Szabó, Z., Bordered knot algebras with matchings , Quantum Topol. 10(3) (2019), 481–592.CrossRefGoogle Scholar
Petkova, I. and Vértesi, V., Combinatorial tangle Floer homology , Geom. Topol. 20(6) (2016), 3219–3332.CrossRefGoogle Scholar
Rasmussen, J., Floer homology and knot complements, PhD Thesis, Harvard University, 2003.Google Scholar
Rasmussen, J., “ Knot polynomials and knot homologies ”, in Geometry and topology of manifolds, Fields Inst. Commun. 47 , American Mathematical Society, Providence, RI, 2005, 261–280.Google Scholar
Zarev, R., Bordered sutured Floer homology, Ph.D. dissertation, Columbia University, 2011.Google Scholar