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Bordered knot algebras with matchings. (English) Zbl 1439.57032

Summary: This paper generalizes the bordered-algebraic knot invariant introduced in an earlier paper [P. Ozsváth and Z. Szabó, Adv. Math. 328, 1088–1198 (2018; Zbl 1417.57015)], giving an invariant now with more algebraic structure. It also introduces signs to define these invariants with integral coefficients. We describe effective computations of the resulting invariant.

MSC:

57K18 Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.)
57R58 Floer homology

Citations:

Zbl 1417.57015

Software:

HFKcalc

References:

[1] J. Hom, Bordered Heegaard Floer homology and the tau-invariant of cable knots. J. Topol. 7(2014), no. 2, 287-326.MR 3217622 Zbl 1368.57002 Bordered knot algebras with matchings591 · Zbl 1368.57002
[2] J. Hom, The knot Floer complex and the smooth concordance group. Comment. Math. Helv. 89(2014), no. 3, 537-570.MR 3260841 Zbl 1312.57008 · Zbl 1312.57008
[3] L. H. Kauffman, Formal knot theory. Mathematical Notes, 30. Princeton University Press, Princeton, N.J., 1983.MR 0712133 Zbl 0537.57002
[4] B. Keller, A-infinity algebras, modules and functor categories. In J. A. de la Peña and R. Bautista (eds.), Trends in representation theory of algebras and related topics. Papers from the Workshop on Representations of Algebras and Related Topics held in Querétaro, August 11-14, 2004, 67-93.MR 2258042 Zbl 1121.18008 · Zbl 1121.18008
[5] M. Khovanov, A functor-valued invariant of tangles. Algebr. Geom. Topol. 2 (2002), 665-741.MR 1928174 Zbl 1002.57006 · Zbl 1002.57006
[6] M. Khovanov and P. Seidel, Quivers, Floer cohomology, and braid group actions. J. Amer. Math. Soc. 15(2002), no. 1, 203-271.MR 1862802 Zbl 1035.53122 · Zbl 1035.53122
[7] R. Lipshitz, P. S. Ozsváth, and D. P. Thurston, A bordered HFalgebra for the torus. In preparation. · Zbl 1201.57025
[8] R. Lipshitz, P. S. Ozsváth, and D. P. Thurston, ComputingHF by factoring mappingb classes. Geom. Topol. 18 (2014), no. 5, 2547-2681.MR 3285222 Zbl 1320.57018 · Zbl 1320.57018
[9] R. Lipshitz, P. S. Ozsváth, and D. P. Thurston, Bimodules in bordered Heegaard Floer homology. Geom. Topol. 19 (2015), no. 2, 525-724.MR 3336273 Zbl 1315.57036 · Zbl 1315.57036
[10] A. Manion, Khovanov-Seidel quiver algebras and Ozsváth-Szabó’s bordered theory. J. Algebra 488(2017), 110-144.MR 3680914 Zbl 1409.16010 · Zbl 1409.16010
[11] P. S. Ozsváth, A. I. Stipsicz, and Z. Szabó, Grid homology for knots and links. Mathematical Surveys and Monographs, 208. American Mathematical Society, Providence, R.I., 2015.MR 3381987 Zbl 1348.57002 · Zbl 1348.57002
[12] P. S. Ozsváth and Z. Szabó, Algebras with matchings and knot Floer homology. In preparation. · Zbl 1052.57012
[13] P. S. Ozsváth and Z. Szabó, Knot Floer homology calculator. https://www.math.princeton.edu/ szabo/HFKcalc.html · Zbl 1448.57015
[14] P. S. Ozsváth and Z. Szabó, The pong algebra. In preparation. · Zbl 0967.53052
[15] P. S. Ozsváth and Z. Szabó, Knot Floer homology and the four-ball genus. Geom. Topol. 7(2003), 615-639.MR 2026543 Zbl 1037.57027 · Zbl 1037.57027
[16] P. S. Ozsváth and Z. Szabó, Holomorphic disks and knot invariants. Adv. Math. 186 (2004), no. 1, 58-116.MR 2065507 Zbl 1062.57019 · Zbl 1062.57019
[17] P. S. Ozsváth and Z. Szabó, Knot Floer homology and integer surgeries. Algebr. Geom. Topol. 8(2008), no. 1, 101-153.MR 2377279 Zbl 1181.57018 · Zbl 1181.57018
[18] P. S. Ozsváth and Z. Szabó, Knot Floer homology and rational surgeries. Algebr. Geom. Topol. 11(2011), no. 1, 1-68.MR 2764036 Zbl 1226.57044 · Zbl 1226.57044
[19] P. S. Ozsváth and Z. Szabó, Kauffman states, bordered algebras, and a bigraded knot invariant. Adv. Math. 328 (2018), 1088-1198.MR 3771149 Zbl 06850703 592P. S. Ozsváth and Z. Szabó · Zbl 1417.57015
[20] I. Petkova and V. Vértesi, Combinatorial tangle Floer homology. Geom. Topol. 20 (2016), no. 6, 3219-3332.MR 3590353 Zbl 1366.57005 · Zbl 1366.57005
[21] J. Rasmussen, Khovanov homology and the slice genus. Invent. Math. 182 (2010), no. 2, 419-447.MR 2729272 Zbl 1211.57009 · Zbl 1211.57009
[22] J. A. Rasmussen, Floer homology and knot complements. Ph.D. thesis. Harvard University, Cambridge, MA, 2003.MR 2704683 arXiv:math/0306378[math.GT]
[23] S. Sarkar, Grid diagrams and the Ozsváth-Szabó tau-invariant. Math. Res. Lett. 18 (2011), no. 6, 1239-1257.MR 2915478 Zbl 1290.57017 · Zbl 1290.57017
[24] R. Zarev, Bordered Floer homology for sutured manifolds. To appear in Geom. Topol. Preprint, 2009.arXiv:0908.1106[math.GT]
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