No file available
To read the file of this research,
you can request a copy directly from the author.
Link homology theories from symplectic geometry
Preprints and early-stage research may not have been peer reviewed yet.
Abstract
For each positive integer n, Khovanov and Rozansky constructed an invariant of links in the form of a doubly-graded cohomology theory whose Euler characteristic is the sl(n) link polynomial. We use Lagrangian Floer cohomology on some suitable affine varieties to build a similar series of link invariants, and we conjecture them to be equal to those of Khovanov and Rozansky after a collapsation of the bigrading. Our work is a generalization of that of Seidel and Smith, who treated the case n=2.
ResearchGate has not been able to resolve any citations for this publication.
The generalization of Jones polynomial of links to the case of graphs inR
3 is presented. It is constructed as the functor from the category of graphs to the category of representations of the quantum groups.
We construct a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial.
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 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.
We define an invariant of oriented links in S3 using the symplectic geometry of certain spaces that arise naturally in Lie theory. More specifically, we present a knot as the closure of a braid that, in turn, we view as a loop in configuration space. Fix an affine subspace script S sign m of the Lie algebra script s sign script l sign2m(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 script S m gives rise to a symplectic fibre bundle over configuration space. An inductive argument constructs a distinguished Lagrangian submanifold L ℘± of a fibre script Y signm, t0 of this fibre bundle; we regard the braid β as a symplectic automorphism of the fibre and apply Lagrangian Floer cohomology to L℘± and β(L℘±) inside script Y signm,t0. The main theorem asserts that this group is invariant under the Markov moves and hence defines an oriented link invariant. We conjecture that this invariant coincides with Khovanov's combinatorially defined link homology theory, after collapsing the bigrading of the latter to a single grading.
We construct a family of rings. To a plane diagram of a tangle we associate a complex of bimodules over these rings. Chain homotopy equivalence class of this complex is an invariant of the tangle. On the level of Grothendieck groups this invariant descends to the Kauffman bracket of the tangle. When the tangle is a link, the invariant specializes to the bigraded cohomology theory introduced in our earlier work.
Let P be a compact symplectic manifold and let L ⊂ P be a Lagrangian submanifold with π2(P, L) = 0. For any exact diffeomorphism ϕ of P with the property that ϕ(L) intersects L transverally, we prove a Morse inequality relating the set ϕ(L) ∩ L to the cohomology of L. As a consequence, we prove a special case of the Arnold conjecture: If π22 (P) = 0 and ϕ is an exact diffeomorphism all of whose fixed points are nondegenerate, then the number of fixed points is greater than or equal to the sum over the Z2-Betti numbers of P. © 1988, International Press of Boston, Inc. All Rights Reserved.
The purpose of this paper is to present a certain combinatorial method of constructing invariants of isotopy classes of oriented tame links. This arises as a generalization of the known polynomial invariants of Conway and Jones. These invariants have one striking common feature. If L+, L- and L0 are diagrams of oriented links which are identical, except near one crossing point (as in Conway or Jones polynomials), then an invariant w(L) has the property: w(L+) is uniquely determined by w(L-) and w(L0), and also w(L-) is uniquely determined by w(L+) and w(L0). To formalize this property we introduce a concept of a Conway algebra and quasi Conway algebra. The paper is now almost 32 years old and it contains the PT contribution to HOMFLYPT polynomial. We believe it should be available in arXiv.
The paper develops a Picard-Lefschetz theory for the coadjoint quotient of a semisimple Lie algebra and analyzes the resulting monodromy representation of the Weyl group.
We define a Floer-homology invariant for knots in an oriented three-manifold, closely related to the Heegaard Floer homologies for three-manifolds defined in an earlier paper. We set up basic properties of these invariants, including an Euler characteristic calculation, and a description of the behavior under connected sums. Then, we establish a relationship with HF+ for surgeries along the knot. Applications include calculation of HF+ of three-manifolds obtained by surgeries on some special knots in S3, and also calculation of HF+ for certain simple three-manifolds which fiber over the circle.
We use the Ozsvath-Szabo theory of Floer homology to define an invariant of knot complements in three-manifolds. This invariant takes the form of a filtered chain complex, which we call CF_r. It carries information about the Floer homology of large integral surgeries on the knot. Using the exact triangle, we derive information about other surgeries on knots, and about the maps on Floer homology induced by certain surgery cobordisms. We define a certain class of \em{perfect} knots in S^3 for which CF_r has a particularly simple form. For these knots, formal properties of the Ozsvath-Szabo theory enable us to make a complete calculation of the Floer homology. This is the author's thesis; many of the results have been independently discovered by Ozsvath and Szabo in math.GT/0209056.
In the usual setup, the grading on Floer homology is relative: it is unique only up to adding a constant. "Graded Lagrangian submanifolds" are Lagrangian submanifolds with a bit of extra structure, which fixes the ambiguity in the grading. The idea is originally due to Kontsevich. This paper contains an exposition of the theory. Several applications are given, amongst them: (1) topological restrictions on Lagrangian submanifolds of projective space, (2) the existence of "symplectically knotted" Lagrangian spheres on a K3 surface, (3) a result about the symplectic monodromy of weighted homogeneous hypersurface singularities. Revised version: minor modifications, journal reference added.
We propose a framework for unifying the sl(N) Khovanov– Rozansky homology (for all N) with the knot Floer homology. We argue that this unification should be accomplished by a triply graded homology theory that categorifies the HOMFLY polynomial. Moreover, this theory should have an additional formal structure of a family of differentials. Roughly speaking, the triply graded theory by itself captures the large-N behavior of the sl(N) homology, and differentials capture nonstable behavior for small N, including knot Floer homology. The differentials themselves should come from another variant of sl(N) homology, namely the deformations of it studied by Gornik, building on work of Lee.
While we do not give a mathematical definition of the triply graded theory, the rich formal structure we propose is powerful enough to make many nontrivial predictions about the existing knot homologies that can then be checked directly. We include many examples in which we can exhibit a likely candidate for the triply graded theory, and these demonstrate the internal consistency of our axioms. We conclude with a detailed study of torus knots, developing a picture that gives new predictions even for the original sl(2) Khovanov homology.
We give a positive answer to the Berry-Robbins problem for any compact Lie group G, i.e. we show the existence of a smooth W-equivariant map from the space of regular triples in a Cartan subalgebra to the flag manifold G/T. This map is constructed via solutions to Nahm's equations and it is compatible with the SO(3) action, where SO(3) acts on G/T via a regular homomorphism from SU(2) to G. We then generalize this picture to include an arbitrary homomorphism from SU(2) to G. This leads to an interesting geometrical picture which appears to be related to the Springer representation of the Weyl group and the work of Kazhdan and Lusztig on representations of Hecke algebras.
The aim of this article is to introduce and study certain topological invariants for closed, oriented three-manifolds Y. These groups are relatively Z-graded Abelian groups associated to SpinC structures over Y. Given a genus g Heegaard splitting of Y, these theories are variants of the Lagrangian Floer homology for the g-fold symmetric product of the surface relative to certain totally real subspaces associated to the handlebodies. Comment: 118 pages, 10 figures, to appear in Annals of Mathematics. Reorganized both this paper and its sequel: the first paper now gives the definitions for closed, oriented three-manifolds. Properties and examples are given in the second paper
- W Borho
- R Macpherson
W. Borho and R. MacPherson, Partial resolutions of nilpotent varieties, Astérisque 101-102 (1983),
23-74.
Homological algebra of mirror symmetry
- M Kontsevich
M. Kontsevich, Homological algebra of mirror symmetry, in Proceedings of the International Congress of Mathematicians (Zürich, 1994), p. 120-139, Birkhäuser, 1995.
Novikov rings and clean intersections, in Northern California Symplectic Geometry Seminar
- M Pozniak
- Floer Homology
M. Pozniak, Floer homology, Novikov rings and clean intersections, in Northern California Symplectic Geometry Seminar, p. 119-181, Amer. Math. Soc., 1999.