Sutured Khovanov homology, Hochschild homology, and the Ozsváth-Szabó spectral sequence

Auroux, Denis; Grigsby, J.; Wehrli, Stephan

doi:10.1090/S0002-9947-2015-06252-7

Sutured Khovanov homology, Hochschild homology, and the Ozsváth-Szabó spectral sequence
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by Denis Auroux, J. Elisenda Grigsby and Stephan M. Wehrli PDF

Trans. Amer. Math. Soc. 367 (2015), 7103-7131 Request permission

Abstract:

In 2002, Khovanov-Seidel constructed a faithful action of the $(m+1)$–strand braid group, $\mathfrak {B}_{m+1}$, on the derived category of left modules over a quiver algebra, $A_m$. We interpret the Hochschild homology of the Khovanov-Seidel braid invariant as a direct summand of the sutured Khovanov homology of the annular braid closure.

References

Marta M. Asaeda, Józef H. Przytycki, and Adam S. Sikora, Categorification of the Kauffman bracket skein module of $I$-bundles over surfaces, Algebr. Geom. Topol. 4 (2004), 1177–1210. MR 2113902, DOI 10.2140/agt.2004.4.1177
Denis Auroux, Fukaya categories of symmetric products and bordered Heegaard-Floer homology, J. Gökova Geom. Topol. GGT 4 (2010), 1–54. MR 2755992
Denis Auroux, J. Elisenda Grigsby, and Stephan M. Wehrli, Khovanov-Seidel quiver algebras and bordered Floer homology, Selecta Math. (N.S.) 20 (2014), no. 1, 1–55. MR 3147412, DOI 10.1007/s00029-012-0106-2
John A. Baldwin and J. Elisenda Grigsby, Categorified invariants and the braid group, Proc. Amer. Math. Soc., electronically published February 26, 2015, DOI 10.1090/S0002-9939-2015-12482-3.
Jonathan Brundan and Catharina Stroppel, Highest weight categories arising from Khovanov’s diagram algebra I: cellularity, Mosc. Math. J. 11 (2011), no. 4, 685–722, 821–822 (English, with English and Russian summaries). MR 2918294, DOI 10.17323/1609-4514-2011-11-4-685-722
Yanfeng Chen and Mikhail Khovanov, An invariant of tangle cobordisms via subquotients of arc rings, Fund. Math. 225 (2014), no. 1, 23–44. MR 3205563, DOI 10.4064/fm225-1-2
Igor Frenkel, Catharina Stroppel, and Joshua Sussan, Categorifying fractional Euler characteristics, Jones-Wenzl projectors and $3j$-symbols, Quantum Topol. 3 (2012), no. 2, 181–253. MR 2901970, DOI 10.4171/QT/28
J. Elisenda Grigsby and Stephan M. Wehrli, Khovanov homology, sutured Floer homology and annular links, Algebr. Geom. Topol. 10 (2010), no. 4, 2009–2039. MR 2728482, DOI 10.2140/agt.2010.10.2009
J. Elisenda Grigsby and Stephan M. Wehrli, On gradings in Khovanov homology and sutured Floer homology, Topology and geometry in dimension three, Contemp. Math., vol. 560, Amer. Math. Soc., Providence, RI, 2011, pp. 111–128. MR 2866927, DOI 10.1090/conm/560/11095
András Juhász, Holomorphic discs and sutured manifolds, Algebr. Geom. Topol. 6 (2006), 1429–1457. MR 2253454, DOI 10.2140/agt.2006.6.1429
András Juhász, Floer homology and surface decompositions, Geom. Topol. 12 (2008), no. 1, 299–350. MR 2390347, DOI 10.2140/gt.2008.12.299
Bernhard Keller, Introduction to $A$-infinity algebras and modules, Homology Homotopy Appl. 3 (2001), no. 1, 1–35. MR 1854636, DOI 10.4310/hha.2001.v3.n1.a1
Bernhard Keller, $A$-infinity algebras, modules and functor categories, Trends in representation theory of algebras and related topics, Contemp. Math., vol. 406, Amer. Math. Soc., Providence, RI, 2006, pp. 67–93. MR 2258042, DOI 10.1090/conm/406/07654
Mikhail Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359–426. MR 1740682, DOI 10.1215/S0012-7094-00-10131-7
Mikhail Khovanov, A functor-valued invariant of tangles, Algebr. Geom. Topol. 2 (2002), 665–741. MR 1928174, DOI 10.2140/agt.2002.2.665
Mikhail Khovanov and Paul Seidel, Quivers, Floer cohomology, and braid group actions, J. Amer. Math. Soc. 15 (2002), no. 1, 203–271. MR 1862802, DOI 10.1090/S0894-0347-01-00374-5
Kenji Lefèvre-Hasegawa, Sur les $A_{\infty }$-catégories, Ph.D. thesis, Université Paris 7, 2003, math.CT/0310337.
Robert Lipshitz, Peter Ozsváth, and Dylan Thurston, Bordered Heegaard Floer homology: Invariance and pairing, math.GT/0810.0687, 2008.
Robert Lipshitz, Peter S. Ozsváth, and Dylan P. Thurston, Bimodules in bordered Heegaard Floer homology, math.GT/1003.0598, 2010. Geom. Topol., to appear.
Ciprian Manolescu, Peter Ozsváth, and Sucharit Sarkar, A combinatorial description of knot Floer homology, Ann. of Math. (2) 169 (2009), no. 2, 633–660. MR 2480614, DOI 10.4007/annals.2009.169.633
Peter Ozsváth and Zoltán Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004), no. 1, 58–116. MR 2065507, DOI 10.1016/j.aim.2003.05.001
Peter Ozsváth and Zoltán Szabó, On the Heegaard Floer homology of branched double-covers, Adv. Math. 194 (2005), no. 1, 1–33. MR 2141852, DOI 10.1016/j.aim.2004.05.008
Jacob Andrew Rasmussen, Floer homology and knot complements, ProQuest LLC, Ann Arbor, MI, 2003. Thesis (Ph.D.)–Harvard University. MR 2704683
Lawrence P. Roberts, On knot Floer homology in double branched covers, Geom. Topol. 17 (2013), no. 1, 413–467. MR 3035332, DOI 10.2140/gt.2013.17.413
Paul Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR 2441780, DOI 10.4171/063
Catharina Stroppel, Parabolic category $\scr O$, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology, Compos. Math. 145 (2009), no. 4, 954–992. MR 2521250, DOI 10.1112/S0010437X09004035
Catharina Stroppel and Ben Webster, 2-block Springer fibers: convolution algebras and coherent sheaves, Comment. Math. Helv. 87 (2012), no. 2, 477–520. MR 2914857, DOI 10.4171/CMH/261
Rumen Zarev, Bordered Floer homology for sutured manifolds, math.GT/0908.1106, 2009.
—, Joining and gluing sutured Floer homology, math.GT/1010.3496, 2010.