A link invariant from the symplectic geometry of nilpotent slices

By P. Seidel, I. Smith. 2004

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We define an invariant of oriented links in S 3 using the symplectic geometry of certain spaces which arise naturally in Lie theory. More specifically, we present a knot as the closure of a braid, which in turn we view as a loop in configuration space. Fix an affine subspaceSm of the Lie algebra sl2m(C) which is a transverse slice to the adjoint action at a nilpotent matrix with two equal Jordan blocks. The adjoint quotient map restricted to Sm gives rise to a symplectic fibre bundle over configuration space. An inductive argument constructs a distinguished Lagrangian submani

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/72b125e682d45a61afcdc4b87501903aca42a725

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Citations in the corpus

The Superpolynomial for Knot Homologies (2005, unknown): “The Superpolynomial for Knot Homologies Nathan M. Dunfield, Sergei Gukov, and Jacob Rasmussen CONTENTS”
Braid cobordisms, triangulated categories, and flag varieties (2006, background): “ sln,and on its subcategories [St2]. • Actions on categories of complexes of matrix factorizations [KR]. • Actions on Fukaya-Floer categories of various symplectic manifolds, such as quiver varieties [SS]. • Actions on derived categories of coherent sheaves on quiver varieties and other Calabi-Yau m”
Localization for Involutions in Floer Cohomology (2010, methodology): “A variant of the argument at the end of [27] shows that”
Grid Homology for Knots and Links (2015, background): “Note that Khovanov and Khovanov-Rozansky have constructed other homology theories for knots [103, 104, 105] that satisfy similar skein exact sequences; compare also [131, 211].”
Floer cohomology and geometric composition of Lagrangian correspondences (2009, unknown): “13 68 v2 [ m at h. SG ] 1 3 A ug 2”
2-block Springer fibers: convolution algebras and coherent sheaves (2008, methodology): “These spaces were for instance used in the geometric construction of knot invariants via Floer homology in the work of [ SS06 ] and [Man07].”
Nilpotent slices, Hilbert schemes, and the Jones polynomial (2004, unknown): “In this section we recall the definitions of the manifold Ym,τ appearing in the Seidel-Smith construction, of the Hilbert scheme Hilbn(Sτ ), and then we prove Theorem 1.”
Khovanov homology from Floer cohomology (2015, methodology): “(In [50] we considered the codimension one subspace lying in sl2n(C) and the adjoint quotient on configurations of total mass zero, but the results carry overmutatis mutandis without the trace zero condition.”
Fivebranes and Knots (2011, methodology): “In one construction [21,22], the effective theory on Σ seems to be a Landau-Ginzburg B-model (so that the branes are matrix factorizations); in a second construction [23], the effective theory is a B-model with target space a certain Kahler manifold; other approaches [24, 25] are based on A-models.”
Knot homology groups from instantons (2008, result): “The development of these ideas was strongly influenced by the paper of Seidel and Smith [40].”
Knot homology via derived categories of coherent sheaves II, $\mathfrak{sl}_{m}$ case (2007, methodology): “ruction. We can now explain one motivation for our work. In his paper [Ma], Manolescu gives a construction of a link invariant using symplectic geometry, generalizing the construction of Seidel-Smith [SS]. Following [Ma], consider the subgroup Σk×Σk ⊂ Σ2k, where Σn denotes the symmetric group, and c”
A Legendrian Thurston-Bennequin bound from Khovanov homology (2005, background): “If the two strands emanating from the right of p meet at a right cusp (i.e., p is part of a “fish”), then the desingularization of F is the same as the desingularization of Res0(F, p) except for the addition of a negative kink.”
A link-splitting spectral sequence in Khovanov homology (2013, background): “In [25], Seidel and Smith associate to a braid β a pair of Lagrangians in an exact symplectic manifold Yn,t0 .”
Quantizations of conical symplectic resolutions I: local and global structure (2012, methodology): “kaya category of M (Remark 4.2). In this interpretation, we expect the action in Conjecture 6.35 to be given by parallel transport in the universal deformation, along the lines of the construction in [SS06] for Slodowy slices of type A. Remark 6.37 As in Section 6.1, we may replace D(Aλ -mod) in the”