Nilpotent slices, Hilbert schemes, and the Jones polynomial

By Ciprian Manolescu. 2004

68 citations recorded by the source corpus.

Seidel and Smith have constructed an invariant of links as the Floer cohomology for two Lagrangians inside a complex affine variety Y. This variety is the intersection of a semisimple orbit with a transverse slice at a nilpotent in the Lie algebra $sl_{2m}.$ We exhibit bijections between a set of generators for the Seidel-Smith cochain complex, the generators in Bigelow's picture of the Jones polynomial, and the generators of the Heegaard Floer cochain complex for the double branched cover. This is done by presenting Y as an open subset of the Hilbert scheme of a Milnor fiber.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/61c310da18b2c9d2875006dbc8e41f7e5896491a

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In Discrete Mathematics and Combinatorics

1. I. Smith
2. M. Abouzaid
3. P. Seidel
4. Lenhard L. Ng
5. Richard Thomas
6. Mikhail Khovanov
7. Ben Webster
8. E. Witten

In Geometry and Topology

1. I. Smith
2. M. Abouzaid
3. P. Seidel
4. T. Ekholm
5. D. Auroux
6. Richard Thomas
7. Lenhard L. Ng
8. Mikhail Khovanov

Citations in the corpus

• Braid cobordisms, triangulated categories, and flag varieties (2006, background): “gory of T∗Flabove, by the deformation of adjoint orbits into the cotangent bundle of the flag variety (by taking the usual simultaneous resolution of the degeneration to the nilpotent cone). Manolescu [Ma1] shows that it is also closely related to the example (3.15), as it sits inside the Hilbert sch”
• Localization for Involutions in Floer Cohomology (2010, methodology): “whose image is the open subset of those subschemes whose image under projection x : X→ C is again of length m. We use slightly different coordinates on X than [ Ma , §2.4], but it is straightforward to adapt the explicit formulae given there.”
• Khovanov homology from Floer cohomology (2015, methodology): “ the nilpotent slice with the form −ddcψin such a way that the Lagrangians obtained as products of matching spheres are Hamiltonian isotopic to the iterated vanishing cycles, by a result of Manolescu [29]. The K¨ahler form from section 5.1 was only used to simplify the study of cohomological propert”
• A Legendrian Thurston-Bennequin bound from Khovanov homology (2005, background): “Thus HKh ∗ sh(F ) = HKh∗−1sh (Res0(F, p)) for all ∗, and the induction assumption that Res0(F, p) is −c(F )-vanishing shows that F is as well.”
• The Ozsváth-Szabó and Rasmussen Concordance Invariants are not Equal (2005, unknown): “(3) Strongly quasipositive knots, in particular positive knots: s(K) = 2τ (K) = 2g(K).”