×

Khovanov-Seidel quiver algebras and bordered Floer homology. (English) Zbl 1296.57012

Let \(\sigma \subset D^2 \times I\) be a braid, and consider the closure of the braid in the solid torus viewed as a product sutured annulus, \(A \times I\). In a work in progress [Sutured Khovanov homology, Hochschild homology, and the Ozsváth-Szabó spectral sequence, {Kh\) associated to a marked disk \(D_m\) equipped with a specific basis of curves and a module \({\mathcal M}_\sigma^{Kh}\) associated to each braid \(\sigma\). Section 4 describes the analogous bordered Floer algebra \(B^{HF}\) and bimodule \({\mathcal M}^{HF}_\sigma\), as well as the 1-moving strands algebra \({\mathcal A( \mathcal Z_{\mathcal Q}},1)\).
In Section 5 the authors use the basis \(\tilde{\mathcal Q}\) from the previous sections to establish the existence of a spectral sequence connecting \(B^{Kh}\) to \(B^{HF}\).
{ Theorem 5.1} Let \[ B^{Kh} := H_\ast \left(\bigoplus_{i,j = 0}^m Hom_A(Q_i,Q_j) \right) \] be the homology of the Hom algebra associated to the basis \(\tilde{\mathcal Q}\) and let \(B^{HF} := {\mathcal A( \mathcal Z_{\mathcal Q}},1)\) be the 1-moving strands algebra associated to the arc diagram, \({\mathcal Z_{\mathcal Q}}\). There exists a filtration on \(B^{HF}\) whose associated graded algebra is isomorphic, as an ungraded algebra, to \(B^{Kh}\). Accordingly, one obtains a spectral sequence whose \(E^1\) page is isomorphic to \(B^{Kh}\) and whose \(E^\infty\) page is isomorphic to \(B^{HF}\).
In Section 6 the authors prove the analogous result for the modules \({\mathcal M}_\sigma^{Kh}\) and \({\mathcal M}^{HF}_\sigma\).
{ Theorem 6.1} Let \(\sigma \in B_{m+1}\) be a braid, \({\mathcal M}_\sigma^{Kh}\) the bimodule associated to the pair \((\tilde{\mathcal Q},\sigma)\) in Sect. 3, and \({\mathcal M}^{HF}_\sigma\) the bordered Floer bimodule associated to the pair \(({\mathcal Q},\sigma)\) in Sect. 4. There exists a filtration on \({\mathcal M}^{HF}_\sigma\) whose associated graded bimodule is isomorphic (as an ungraded \(A_\infty\) bimodule over \(B^{Kh}\)) to \({\mathcal M}_\sigma^{Kh}\). Accordingly, one obtains a spectral sequence whose \(E^1\) page is isomorphic to \({\mathcal M}_\sigma^{Kh}\) and whose \(E^\infty\) page is isomorphic to \({\mathcal M}_\sigma^{HF}\).
Section 7 contains an example with a 3-braid whose closure is the positive \((3,5)\) torus knot \(T_{3,5}\).

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R58 Floer homology
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Asaeda, M.M., Przytycki, J.H., Sikora, A.S.: Categorification of the Kauffman bracket skein module of \[I\]-bundles over surfaces. Algebr. Geom. Topol. 4,1177-1210 (2004) [electronic] · Zbl 1070.57008
[2] Auroux, D.: Fukaya categories and bordered Heegaard-Floer homology. In: Proceedings of International Congress of Mathematicians, vol. II (Hyderabad, 2010), pp. 917-941. Hindustan Book Agency, New Delhi, (2010). math.GT/1003.2962 · Zbl 1275.53082
[3] Auroux, D.: Fukaya categories of symmetric products and bordered Heegaard-Floer homology. J. Gökova Geom. Topol. GGT 4, 1-54, (2010). math.GT/1001.4323 · Zbl 1285.53077
[4] Auroux, D., Grigsby, J.E., Wehrli, S.M.: Sutured Khovanov homology, Hochschild homology, and the Ozsváth-Szabó spectral sequence (in preparation) · Zbl 1365.57011
[5] Baldwin, J.A.: Heegaard Floer homology and genus one, one boundary component open books. Topology 1(4), 963-992 (2008) · Zbl 1160.57009
[6] Baldwin, J.A.: On the spectral sequence from Khovanov homology to Heegaard Floer homology. Int. Math. Res. Not. IMRN 15, 3426-3470 (2011) · Zbl 1231.57019
[7] Baldwin, J.A., Plamenevskaya, O.: Khovanov homology, open books, and tight contact structures. Adv. Math. 224(6), 2544-2582 (2010) · Zbl 1239.57033
[8] Berglund, A.: A-infinity algebras and homological perturbation theory (2010)
[9] Bernstein, J., Lunts, V.: Equivariant Sheaves and Functors. Lecture Notes in Mathematics, vol. 1578 Springer, Berlin (1994) · Zbl 0808.14038
[10] Brundan, J., Stroppel, C.: Highest weight categories arising from Khovanov’s diagram algebra I: cellularity. Mosc. Math. J. 11(4), 685-722 (2011) · Zbl 1275.17012
[11] Chen, Y., Khovanov, M.: An invariant of tangle cobordisms via subquotients of arc rings (2006). math.QA/0610054 · Zbl 1321.57031
[12] Grigsby, J.E., Wehrli, S.M.: Khovanov homology, sutured Floer homology and annular links. Algebr. Geom. Topol 10(4), 2009-2039 (2010) · Zbl 1206.57012 · doi:10.2140/agt.2010.10.2009
[13] Grigsby, J.E., Wehrli, S.M.: On the colored Jones polynomial, sutured Floer homology, and knot Floer homology. Adv. Math. 223(6), 2114-2165 (2010) · Zbl 1205.57015 · doi:10.1016/j.aim.2009.11.002
[14] Grigsby, J.E., Wehrli, S.M.: On the naturality of the spectral sequence from Khovanov homology to Heegaard Floer homology. Int. Math. Res. Not. IMRN (21), 4159-4210 (2010) · Zbl 1230.57012
[15] Grigsby, J.E., Wehrli, S.M.: On gradings in Khovanov homology and sutured Floer homology. In: Topology and Geometry in Dimension Three. Contemporary Mathematics, vol. 560, pp. 111-128. American Mathematical Society, Providence (2011) · Zbl 1333.57023
[16] Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002) · Zbl 1044.55001
[17] Juhász, A.: Holomorphic discs and sutured manifolds. Algebr. Geom. Topol. 6, 1429-1457 (2006) [electronic] · Zbl 1129.57039
[18] Kadeishvili, T.V.: The algebraic structure in the homology of an \[A(\infty )\]-algebra. Soobshch. Akad. Nauk Gruzin. SSR 108(2), 249-252 (1983). 1982 · Zbl 0535.55005
[19] Keller, B.: Introduction to \[A\]-infinity algebras and modules. Homol. Homot. Appl. 3(1), 1-35 (2001) · Zbl 0989.18009
[20] Keller, \[B.: A\]-infinity algebras, modules and functor categories. In: Trends in Representation Theory of Algebras and Related Topics. Contemporary Mathematics, vol. 406, pp. 67-93. American Mathematical Society, Providence (2006) · Zbl 1121.18008
[21] Khovanov, M.: A categorification of the Jones polynomial. Duke Math. J. 101(3), 359-426 (2000) · Zbl 0960.57005 · doi:10.1215/S0012-7094-00-10131-7
[22] Khovanov, M.: A functor-valued invariant of tangles. Alg. Geom. Topol. 2, 665-741 (2002) · Zbl 1002.57006 · doi:10.2140/agt.2002.2.665
[23] Khovanov, M., Seidel, P.: Quivers, Floer cohomology, and braid group actions. J. Am. Math. Soc. 15(1), 203-271 (2002) [electronic] · Zbl 1035.53122
[24] Klamt, A.: \[A_\infty\] Structures on the Algebra of Extensions of Verma Modules in the Parabolic Category O. Diploma Thesis, Rheinischen Friedrich-Wilhelms-Universität Bonn (2010). math.RT/1104.0102
[25] Klamt, A., Stroppel, C.: On the Ext algebras of parabolic Verma modules and \[A_\infty \]-structures. J. Pure Appl. Algebra 216(2), 323-336 (2012) · Zbl 1277.17006 · doi:10.1016/j.jpaa.2011.06.015
[26] Kontsevich, M.: Homological Algebra of Mirror Symmetry. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2 (Zürich, 1994), pp. 120-139, Birkhäuser, Basel (1995) · Zbl 0846.53021
[27] Kontsevich, M., Soibelman, Y.: Homological mirror symmetry and torus fibrations. In: Symplectic Geometry and Mirror Symmetry (Seoul, 2000), pp. 203-263. World Scientific Publishing, River Edge (2001) · Zbl 1072.14046
[28] Kronheimer, P.B., Mrowka, T.S.: Khovanov homology is an unknot-detector. Publ. Math. Inst. Hautes Études Sci. 113, 97-208 (2011) · Zbl 1241.57017 · doi:10.1007/s10240-010-0030-y
[29] Lefèvre-Hasegawa, K.: Sur les \[A_{\infty }\]-catégories. PhD thesis, Université Paris 7, Paris (2003). math.CT/0310337
[30] Lekili, Y., Perutz, T.: Fukaya categories of the torus and dehn surgery. PNAS 108(2), 8106-8113 (2011) · Zbl 1256.53055 · doi:10.1073/pnas.1018918108
[31] Lipshitz, R., Ozsváth, P., Thurston, D: Bordered Heegaard Floer homology: invariance and pairing (2008). math.GT/0810.0687 · Zbl 1422.57080
[32] Lipshitz, R., Ozsváth, P., Thurston, D.: Bimodules in bordered Heegaard Floer homology (2010). math.GT/1003.0598 · Zbl 1315.57036
[33] Lipshitz, R., Ozsváth, P., Thurston, D.: Heegaard Floer homology as morphism spaces (2010). math.GT/1005.1248 · Zbl 1236.57042
[34] Lipshitz, R., Ozsváth, P.S., Thurston, D.P.: Bordered Floer homology and the spectral sequence of a branched double cover I (2010). math.GT/1011.0499 · Zbl 1318.53101
[35] Lipshitz, R., Ozsváth, P.S., Thurston, D.P.: A faithful linear-categorical action of the mapping class group of a surface with boundary (2010). math.GT/1012.1032 · Zbl 1280.57016
[36] Loi, A., Piergallini, R.: Compact Stein surfaces with boundary as branched covers of \[B^4\]. Invent. Math. 143, 325-248 (2001) · Zbl 0983.32027 · doi:10.1007/s002220000106
[37] Manolescu, C., Ozsváth, P., Sarkar, S.: A combinatorial description of knot Floer homology. Ann. Math. 169(2), 633-660 (2009) · Zbl 1179.57022 · doi:10.4007/annals.2009.169.633
[38] Markl, M.: Transferring \[A_\infty \] (strongly homotopy associative) structures. Rend. Circ. Mat. Palermo (2) Suppl. (79), 139-151 (2006) · Zbl 1112.18007
[39] Merkulov, S.A.: Strong homotopy algebras of a Kähler manifold. Int. Math. Res. Notices (3), 153-164 (1999) · Zbl 0995.32013
[40] Ozsváth, P., Szabó, Z.: Heegaard Floer homology and contact structures. Duke Math. J. 129(1), 39-61 (2005) · Zbl 1083.57042 · doi:10.1215/S0012-7094-04-12912-4
[41] Ozsváth, P., Szabó, Z.: On the Heegaard Floer homology of branched double-covers. Adv. Math. 194(1), 1-33 (2005) · Zbl 1076.57013 · doi:10.1016/j.aim.2004.05.008
[42] Plamenevskaya, O.: Transverse knots and Khovanov homology. Math. Res. Lett. 13(4), 571-586 (2006) · Zbl 1143.57006 · doi:10.4310/MRL.2006.v13.n4.a7
[43] Roberts, L.P.: On knot Floer homology in double branched covers (2007). math.GT/0706.0741 · Zbl 1415.57009
[44] Roberts, L.P.: Notes on the Heegaard-Floer link surgery spectral sequence (2008). math.GT/0808.2817 · Zbl 1083.57042
[45] Seidel, P.: Vanishing cycles and mutation. In: European Congress of Mathematics, Vol. II (Barcelona, 2000), progress in Mathematics, vol. 202, pp. 65-85. Birkhäuser, Basel (2001) · Zbl 1042.53060
[46] Seidel, P.: A long exact sequence for symplectic Floer cohomology. Topology 42(5), 1003-1063 (2003) · Zbl 1032.57035 · doi:10.1016/S0040-9383(02)00028-9
[47] Seidel, P.: Fukaya categories and Picard-Lefschetz theory. In: Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2008) · Zbl 1159.53001
[48] Seidel, P.: Fukaya \[A_\infty\] structures associated to Lefschetz fibrations I (2009). math.SG/0912.3932 · Zbl 1267.53094
[49] Seidel, P., Smith, I.: Localization for involutions in Floer cohomology. Geom. Funct. Anal. 20(6), 1464-1501 (2010) · Zbl 1210.53084 · doi:10.1007/s00039-010-0099-y
[50] Stroppel, C.: Parabolic category \[\cal{O} \], perverse sheaves on Grassmanians, Springer fibers and Khovanov homology. Compos. Math., pp. 954-992 (2009) · Zbl 1187.17004
[51] Szabó, Z.: A geometric spectral sequence in Khovanov homology (2010). math.GT/1010.4252 · Zbl 1344.57008
[52] The Knot Atlas. http://katlas.org/wiki/10_124
[53] Watson, L.: Surgery obstructions from Khovanov homology (2008). math.GT/0807.1341 · Zbl 1280.57002
[54] Zarev R.: Bordered Floer homology for sutured manifolds (2009). math.GT/0908.1106
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.