Browsing by Author "Ekholm, T"

Now showing 1 - 6 of 6

Open Access
A complete knot invariant from contact homology
Ekholm, T; Ng, L; Shende, V
We construct an enhanced version of knot contact homology, and show that we can deduce from it the group ring of the knot group together with the peripheral subgroup. In particular, it completely determines a knot up to smooth isotopy. The enhancement consists of the (fully noncommutative) Legendrian contact homology associated to the union of the conormal torus of the knot and a disjoint cotangent fiber sphere, along with a product on a filtered part of this homology. As a corollary, we obtain a new, holomorphic-curve proof of a result of the third author that the Legendrian isotopy class of the conormal torus is a complete knot invariant. Furthermore, we relate the holomorphic and sheaf approaches via calculations of partially wrapped Floer homology in the spirit of [BEE12].
Open Access
Higher genus knot contact homology and recursion for colored HOMFLY-PT polynomials
(Advances in Theoretical and Mathematical Physics, 2020-01-01) Ekholm, T; Ng, L
We sketch a construction of Legendrian Symplectic Field Theory (SFT) for conormal tori of knots and links. Using large N duality and Witten’s connection between open Gromov–Witten invariants and Chern–Simons gauge theory, we relate the SFT of a link conormal to the colored HOMFLY-PT polynomials of the link. We present an argument that the HOMFLY-PT wave function is determined from SFT by induction on Euler characteristic, and also show how to, more directly, extract its recursion relation by elimination theory applied to finitely many noncommutative equations. The latter can be viewed as the higher genus counterpart of the relation between the augmentation variety and Gromov–Witten disk potentials established in [1] by Aganagic, Vafa, and the authors, and, from this perspective, our results can be seen as an SFT approach to quantizing the augmentation variety
Open Access
Knot contact homology
(Geometry & Topology, 2013-04-30) Ekholm, T; Etnyre, JB; Ng, L; Sullivan, MG
The conormal lift of a link K in ℝ3is a Legendrian submanifold ∧Kin the unit cotangent bundle U*ℝ3of ℝ3with contact structure equal to the kernel of the Liouville form. Knot contact homology, a topological link invariant of K, is defined as the Legendrian homology of ∧K, the homology of a differential graded algebra generated by Reeb chords whose differential counts holomorphic disks in the symplectization R × U*ℝ3with Lagrangian boundary condition R × ∧K. We perform an explicit and complete computation of the Legendrian homology of ∧Kfor arbitrary links K in terms of a braid presentation of K, confirming a conjecture that this invariant agrees with a previously defined combinatorial version of knot contact homology. The computation uses a double degeneration: the braid degenerates toward a multiple cover of the unknot, which in turn degenerates to a point. Under the first degeneration, holomorphic disks converge to gradient flow trees with quantum corrections. The combined degenerations give rise to a new generalization of flow trees called multiscale flow trees. The theory of multiscale flow trees is the key tool in our computation and is already proving to be useful for other computations as well.
Open Access
Knot contact homology, string topology, and the cord algebra
(Journal de l’École polytechnique — Mathématiques, 2017) Cieliebak, K; Ekholm, T; Latschev, J; Ng, L
The conormal Lagrangian LKof a knot K in R3is the submanifold of the cotangent bundle T∗R3consisting of covectors along K that annihilate tangent vectors to K. By intersecting with the unit cotangent bundle S∗R3, one obtains the unit conormal ΛK, and the Legendrian contact homology of ΛKis a knot invariant of K, known as knot contact homology. We define a version of string topology for strings in R3∪ LKand prove that this is isomorphic in degree 0 to knot contact homology. The string topology perspective gives a topological derivation of the cord algebra (also isomorphic to degree 0 knot contact homology) and relates it to the knot group. Together with the isomorphism this gives a new proof that knot contact homology detects the unknot. Our techniques involve a detailed analysis of certain moduli spaces of holomorphic disks in T∗R3with boundary on R3∪ LK.
Open Access
Legendrian contact homology in the boundary of a subcritical Weinstein 4-manifold
(Journal of Differential Geometry, 2015-09) Ekholm, T; Ng, L
We give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in S1× S2or any connected sum #k(S1×S2), viewed as the contact boundary of the Weinstein manifold obtained by attaching 1-handles to the 4-ball. In view of the surgery formula for symplectic homology [5], this gives a combinatorial description of the symplectic homology of any 4-dimensional Weinstein manifold (and of the linearized contact homology of its boundary). We also study examples and discuss the invariance of the Legendrian homology algebra under deformations, from both the combinatorial and the analytical perspectives.
Open Access
Topological strings, D-model, and knot contact homology
(Advances in Theoretical and Mathematical Physics, 2014) Aganagic, M; Ekholm, T; Ng, L; Vafa, C
© 2014 International Press. We study the connection between topological strings and contact homology recently proposed in the context of knot invariants. In particular, we establish the proposed relation between the Gromov- Witten disk amplitudes of a Lagrangian associated to a knot and augmentations of its contact homology algebra. This also implies the equality between the Q-deformed A-polynomial and the augmentation polynomial of knot contact homology (in the irreducible case). We also generalize this relation to the case of links and to higher rank representations for knots. The generalization involves a study of the quantum moduli space of special Lagrangian branes with higher Betti numbers probing the Calabi-Yau. This leads to an extension of SYZ, and a new notion of mirror symmetry, involving higher dimensional mirrors. The mirror theory is a topological string, related to D-modules, which we call the "D-model". In the present setting, the mirror manifold is the augmentation variety of the link. Connecting further to contact geometry, we study intersection properties of branches of the augmentation variety guided by the relation to D-modules. This study leads us to propose concrete geometric constructions of Lagrangian fillings for links. We also relate the augmentation variety with the large N limit of the colored HOMFLY, which we conjecture to be related to a Qdeformation of the extension of A-polynomials associated with the link complement.