Conormal bundles, contact homology and knot invariants

  title={Conormal bundles, contact homology and knot invariants},
  author={Lenhard L. Ng},
  journal={arXiv: Symplectic Geometry},
  • Lenhard L. Ng
  • Published 16 December 2004
  • Mathematics
  • arXiv: Symplectic Geometry
String theory has provided a beautiful correspondence between enumerative geometryand knot invariants; for details, see the survey by Marino [˜ 16] or other papers in thepresent volume. This correspondence applies methods from physics and algebraicgeometry to a construction, described below, which is essentially symplectic.To symplectic geometers, there is a natural way to study this same construction bycounting holomorphic curves. The symplectic approach leads to a knot invariant whichseems to… 

Figures from this paper

A topological introduction to knot contact homology
Knot contact homology is a Floer-theoretic knot invariant derived from counting holomorphic curves in the cotangent bundle of \(\mathbb{R}^{3}\) with Lagrangian boundary condition on the conormal
Conormal bundles to knots and the Gopakumar-Vafa conjecture
We offer a new construction of Lagrangian submanifolds for the Gopakumar-Vafa conjecture relating the Chern-Simons theory on the 3-sphere and the Gromov-Witten theory on the resolved conifold. Given
Filtrations on the knot contact homology of transverse knots
We construct a new invariant of transverse links in the standard contact structure on $${\mathbb R }^3.$$ This invariant is a doubly filtered version of the knot contact homology differential graded
Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots
We reconsider topological string realization of SU(N) Chern-Simons theory on S^3. At large N, for every knot K in S^3, we obtain a polynomial A_K(x,p;Q) in two variables x,p depending on the t'Hooft
String Topology: Background and Present State
The data of a "2D field theory with a closed string compactification" is an equivariant chain level action of a cell decomposition of the union of all moduli spaces of punctured Riemann surfaces with
Frame ambiguity in Open Gromov-Witten invariants
We consider Open Gromov-Witten invariants for noncompact Calabi-Yau in the case the Lagrangian has the topology of $\R^2 \times S^1$. The definition of the invariant involves the choice of a frame
This is a digest of definitions and results from the minicourse, along with a collection of exercises. Discussion is incomplete to nonexistent, but I hope that the digest will still be useful as a
The homology of path spaces and Floer homology with conormal boundary conditions
Abstract.We define the Floer complex for Hamiltonian orbits on the cotangent bundle of a compact manifold which satisfy non-local conormal boundary conditions. We prove that the homology of this
Knot contact homology
The conormal lift of a link K in ℝ3 is a Legendrian submanifold ΛK in the unit cotangent bundle U∗ℝ3 of ℝ3 with contact structure equal to the kernel of the Liouville form. Knot contact homology, a


Zero-loop open strings in the cotangent bundle and Morse homotopy
0. Introduction. Many important works in symplectic geometry and topology are regarded as the symplectization or the quantization of the corresponding results in ordinary geometry and topology. One
In these lecture notes we discuss a body of work in which Morse theory is used to construct various homology and cohomology operations. In the classical setting of algebraic topology this is done by
The contact homology of Legendrian submanifolds in R2n+1
We define the contact homology for Legendrian submanifolds in standard contact (2n + 1)-space using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex n-space. This
Functors and Computations in Floer Homology with Applications, I
Abstract. This paper is concerned with Floer cohomology of manifolds with contact type boundary. In this case, there is no conjecture on this ring, as opposed to the compact case, where it is
On the Floer homology of cotangent bundles
This paper concerns Floer homology for periodic orbits and for a Lagrangian intersection problem on the cotangent bundle T* M of a compact orientable manifold M. The first result is a new L∞ estimate
On the characteristic and deformation varieties of a knot
The colored Jones function of a knot is a sequence of Laurent polynomials in one variable, whose nth term is the Jones polynomial of the knot colored with the n-dimensional irreducible representation
Invariants of Legendrian Knots and Coherent Orientations
We provide a translation between Chekanov’s combinatorial theory for invariants of Legendrian knots in the standard contact R and a relative version of Eliashberg and Hofer’s contact homology. We use
Framed knot contact homology
We extend knot contact homology to a theory over the ring $\mathbb{Z}[\lambda^{\pm 1},\mu^{\pm 1}]$, with the invariant given topologically and combinatorially. The improved invariant, which is
Plane curves associated to character varieties of 3-manifolds
Consider a compact 3-manifold M with boundary consisting of a single torus. The papers [CS1, CS2, CGLS] discuss the variety of characters of SL2(C) representations of zl(M), and some of the ways in