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The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link. (English) Zbl 1388.14087
The authors formulate a conjecture relating the Poincaré polynomial $$\overline{\mathcal P}(q,a,t)$$ of the triply graded (unreduced) Khovanov-Rozansky HOMFLY homologies of an algebraic link to an other series involving the weight polynomials of certain Hilbert schemes of points supported on the corresponding complex plane curve singularity. Denoting the latter series by $$\overline{\mathcal P}_{\mathrm{alg}}(q,a,t)$$, the conjectured identity $$\overline{\mathcal P}(q,a,t) = \overline{\mathcal P}_{\mathrm{alg}}(q,a,t)$$ setting $$t = -1$$ specializes to an earlier conjecture of A. Oblomkov and V. Shende [Duke Math. J. 161, No. 7, 1277–1303 (2012; Zbl 1256.14025)] relating the HOMFLY polynomial of an algebraic link to a series with coefficients being the Euler characteristics of the above Hilbert scheme spaces associated to the corresponding plane curve singularity.
Several results are proved to support the proposed conjecture. The first is a statement about the structure of $$\overline{\mathcal P}_{\mathrm{alg}}(q,a,t)$$ in terms of the number of branches, the arithmetic genus and the multiplicity of the singularity. A symmetry property with respect to the variable change $$t \mapsto 1/qt$$ is also proved.
It is also discussed that the lowest $$a$$-degree part $$\overline{\mathcal P}_{\mathrm{alg}}^{\min}$$ of $$\overline{\mathcal P}_{\mathrm{alg}}(q,a,t)$$ can be computed from the compactified Jacobian of the singularity.
Next, some explicit formulas are proved for $$\overline{\mathcal P}_{\mathrm{alg}}$$ in the case of algebraic torus knots, through a stratification of the Hilbert scheme type moduli space corresponding to these singularities. The limiting case $$(n,\infty)$$ of the $$(n,k)$$ torus knot results matches the formula for the stable superpolynomial of torus knots conjectured in [N. M. Dunfield et al., Exp. Math. 15, No. 2, 129–159 (2006; Zbl 1118.57012)].
In the $$(n, mn+1)$$ torus knot case, a further conjectural formula (based on a conjecturally identical stratification) is proposed using Cherednik algebras expressing the lowest $$a$$-degree part $$\overline{\mathcal P}_{\mathrm{alg}}^{\min}$$ (and, as explained further in [E. Gorsky et al., Duke Math. J. 163, No. 14, 2709–2794 (2014; Zbl 1318.57010)], even the entire $$\overline{\mathcal P}_{\mathrm{alg}}$$) in terms of an expression involving partitions of $$n$$.
The Appendix by E. Gorsky relates the Hilbert scheme stratification of torus knot type plane curve singularities to the combinatorics of diagonal harmonics and DAHA representations, justifying some conjectures involving $$(q,t)$$-Catalan numbers from E. A. Gorsky [Contemp. Math. 566, 213–232 (2012; Zbl 1294.57007)].

##### MSC:
 14H20 Singularities of curves, local rings 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 57M27 Invariants of knots and $$3$$-manifolds (MSC2010)
##### Keywords:
plane curve; Hilbert scheme; Khovanov homology
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##### References:
 [1] 10.1007/s00029-017-0336-4 · Zbl 1456.57013 [2] ; Altman, Real and complex singularities, 1, (1977) [3] 10.1016/0001-8708(80)90043-2 · Zbl 0427.14015 [4] 10.1007/978-1-4612-3940-6 [5] ; Beĭlinson, Analysis and topology on singular spaces, I. Astérisque, 100, 5, (1982) [6] 10.1215/S0012-7094-03-11824-4 · Zbl 1067.16047 [7] 10.1155/S1073792803210205 · Zbl 1063.20003 [8] 10.1515/crll.2004.020 · Zbl 1067.16046 [9] 10.1007/s00029-009-0507-z · Zbl 1226.20002 [10] ; Bruns, Cohen-Macaulay rings. Cambridge Studies in Advanced Mathematics, 39, (1993) · Zbl 0788.13005 [11] 10.1007/978-0-8176-4745-2_5 · Zbl 1241.32011 [12] 10.1093/imrn/rns202 · Zbl 1329.57019 [13] ; Danilov, Izv. Akad. Nauk SSSR Ser. Mat., 50, 925, (1986) [14] ; Deligne, Actes du Congrès International des Mathématiciens, 425, (1971) [15] 10.1007/BF02684692 · Zbl 0219.14007 [16] 10.1007/BF02685881 · Zbl 0237.14003 [17] ; Deligne, Proceedings of the International Congress of Mathematicians, 79, (1975) [18] ; Dunfield, Experiment. Math., 15, 129, (2006) [19] 10.1007/JHEP03(201 [20] ; Durfee, Geometry and topology. Lecture Notes in Pure and Appl. Math., 105, 99, (1987) [21] ; Egge, Electron. J. Combin., 10, (2003) [22] ; Fantechi, J. Algebraic Geom., 8, 115, (1999) [23] 10.1023/A:1022476211638 · Zbl 0853.05008 [24] 10.1007/s00222-003-0313-8 · Zbl 1071.20005 [25] 10.1016/j.aim.2004.12.005 · Zbl 1084.14005 [26] 10.1215/S0012-7094-06-13213-1 · Zbl 1096.14003 [27] 10.1090/conm/566/11222 · Zbl 1294.57007 [28] 10.1016/j.jcta.2012.07.002 · Zbl 1252.05009 [29] 10.1016/j.matpur.2015.03.003 · Zbl 1349.14012 [30] 10.1215/00127094-2827126 · Zbl 1318.57010 [31] 10.1016/S0001-8708(02)00061-0 · Zbl 1043.05012 [32] 10.1155/S1073792804132509 · Zbl 1069.05075 [33] ; Haglund, The q,t-Catalan numbers and the space of diagonal harmonics. University Lecture Series, 41, (2008) · Zbl 1142.05074 [34] 10.1023/A:1022450120589 · Zbl 0803.13010 [35] 10.1007/s002220200219 · Zbl 1053.14005 [36] 10.1215/kjm/1250520873 · Zbl 0613.14008 [37] 10.1017/S0305004108002016 · Zbl 1183.57006 [38] 10.1142/S0129167X07004400 · Zbl 1124.57003 [39] 10.2140/gt.2008.12.1387 · Zbl 1146.57018 [40] ; Loehr, Electron. J. Combin., 12, (2005) [41] 10.1016/j.jcta.2008.07.001 · Zbl 1188.05012 [42] 10.1016/0040-9383(62)90019-8 · Zbl 0121.38003 [43] 10.1515/crelle-2012-0093 · Zbl 1304.14036 [44] 10.4171/JEMS/423 · Zbl 1303.14019 [45] ; Milnor, Singular points of complex hypersurfaces. Annals of Mathematics Studies, 61, (1968) · Zbl 0184.48405 [46] 10.1007/s10240-010-0026-7 · Zbl 1200.22011 [47] 10.1215/00127094-1593281 · Zbl 1256.14025 [48] 10.1016/j.aim.2016.01.015 · Zbl 1403.20007 [49] 10.1090/S0894-0347-09-00646-8 · Zbl 1250.14035 [50] 10.1007/s00209-006-0021-3 · Zbl 1135.14020 [51] 10.2140/gt.2015.19.3031 · Zbl 1419.57027 [52] 10.2140/agt.2007.7.261 · Zbl 1156.57010 [53] 10.1215/00127094-2009-016 · Zbl 1237.20008 [54] 10.1016/j.aim.2011.05.012 · Zbl 1230.14048
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