Knot contact homology
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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.
SubjectScience & Technology
Published Version (Please cite this version)10.2140/gt.2013.17.975
Publication InfoEkholm, T; Etnyre, JB; Ng, L; & Sullivan, MG (2013). Knot contact homology. Geometry & Topology, 17(2). pp. 975-1112. 10.2140/gt.2013.17.975. Retrieved from https://hdl.handle.net/10161/17788.
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Eads Family Professor
My research mainly focuses on symplectic topology and low-dimensional topology. I am interested in studying structures in symplectic and contact geometry (Weinstein manifolds, contact manifolds, Legendrian and transverse knots), especially through holomorphic-curve techniques. One particular interest is extracting topological information about knots through cotangent bundles, and exploring relations to topological string theory. I have also worked in Heegaard Floer theory, quantum topology, and