Homology of torus (and other) knots: what do we know?
We plan to discuss recent progress in understanding HOMFLY-PT homology of torus links, and related algebro-geometric models.
The main goal is to orient the audience (and the speakers) in the existing literature, and get a clear understanding of the status
of various conjectures.
Picture credit: [DGR]
Contact: Eugene Gorsky
The seminar meets tentatively on Wednesdays at 9-10am US Pacific time= 12-1 US Eastern time = 6pm Central European Time.
3/31 Introduction, definition of HOMFLY homology, y-ification [KhS,GH] notes
4/7 Explicit computations: recursions, full twist [H,HM,GH] notes
4/14 Guest lecture: Anton Mellit notes
4/21 Anton continues notes
4/8 Singular curves: Hilbert schemes, affine Springer fibers [ORS,Hik,GM,GMV2] notes
5/5 Guest lecture: Oscar Kivinen [K] notes
5/12 Guest lecture: Pavel Galashin [GL,GL2] typed notes , handwritten notes .
5/19 Banff workshop
5/26 Braid varieties [MChar,CGGS,CGGS2] notes
6/2 Hilb(C^2): overview [Hai,GNR,OR] notes
Soergel bimodules and HOMFLY-PT homology:
[KhS] M. Khovanov. Triply-graded link homology and Hochschild homology of Soergel bimodules. Internat. J. Math. 18 (2007), no. 8, 869-885.
[DGR] N. Dunfield, S. Gukov, J. Rasmussen. The superpolynomial for knot homologies.
Experiment. Math. 15 (2006), no. 2, 129-159.
[H] M. Hogancamp. Categorified Young symmetrizers and stable homology of torus links. Geom. Topol. 22 (2018), no. 5, 2943-3002.
[HM] M. Hogancamp, A. Mellit. Torus link homology. arXiv:1909.00418
[GH] E. Gorsky, M. Hogancamp. Hilbert schemes and y-ification of Khovanov-Rozansky homology.arXiv:1712.03938
[GHM] E. Gorsky, M. Hogancamp, A. Mellit. Tautological classes and symmetry in Khovanov-Rozansky homology. arXiv:2103.01212
[Hik] T. Hikita. Affine Springer fibers of type A and combinatorics of diagonal coinvariants. Adv. Math. 263 (2014), 88-122.
[ORS] A. Oblomkov, J. Rasmussen, V. Shende. The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link.
Geom. Topol. 22 (2018), no. 2, 645-691.
[MY] D. Maulik, Z. Yun. Macdonald formula for curves with planar singularities. J. Reine Angew. Math. 694 (2014), 27-48.
[OY] A. Oblomkov, Z. Yun. Geometric representations of graded and rational Cherednik algebras. Adv. Math. 292 (2016), 601-706.
[K] O. Kivinen. Unramified affine Springer fibers and isospectral Hilbert schemes. Selecta Math. (N.S.) 26 (2020), no. 4, Paper No. 61, 42 pp.
[GM] E. Gorsky, M. Mazin. Compactified Jacobians and q,t-Catalan numbers, I. J. Combin. Theory Ser. A 120 (2013), no. 1, 49-63.
[GMV] E. Gorsky, M. Mazin, M. Vazirani. Affine permutations and rational slope parking functions. Trans. Amer. Math. Soc. 368 (2016), no. 12, 8403-8445.
[GMV2] E. Gorsky, M. Mazin, M. Vazirani. Recursions for rational q,t-Catalan numbers. J. Combin. Theory Ser. A 173 (2020), 105237, 28 pp.
[MChar] A. Mellit. Cell decompositions of character varieties. arXiv:1905.10685
[GL] P. Galashin, T. Lam. Positroids, knots, and q,t-Catalan numbers. arXiv:2012.09745
[GL2] P. Galashin, T. Lam. Positroid Catalan numbers. arXiv:2104.05701
[CGGS] R. Casals, E. Gorsky, M. Gorsky, J. Simental. Algebraic weaves and braid varieties. arXiv:2012.06931
[CGGS2] R. Casals, E. Gorsky, M. Gorsky, J. Simental. Positroid links and braid varieties. arXiv:2105.13948
Hilbert schemes of C^2:
[Hai] M. Haiman. Hilbert schemes, polygraphs and the Macdonald positivity conjecture. J. Amer. Math. Soc. 14 (2001), no. 4, 941-1006.
[GNR] E. Gorsky, A. Negut, J. Rasmussen. Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology. Adv. Math. 378 (2021), 107542, 115 pp.
[OR] A. Oblomkov, L. Rozansky. Knot homology and sheaves on the Hilbert scheme of points on the plane. Selecta Math. (N.S.) 24 (2018), no. 3,
AIM Research community on link homology
FRG workshop Categorical braid group actions and categorical representation theory.
June 14-18, 2021, University of Massachusetts
Jake Rasmussen's lectures at WARTHOG, 2016.
Problem list from an AIM workshop in 2018
(note: quite a few problems from the list have been solved).
Focused Research Group "Algebra and Geometry Behind Link Homology"