Conormal bundles to knots and the Gopakumar-Vafa conjecture

Details Conormal bundles to knots and the Gopakumar-Vafa conjecture Conormal bundles to knots and the Gopakumar-Vafa conjecture

2005-03-14

arxiv.org/abs/math/0503248v2

Mathematics

Differential Geometry

Proof of the main result simplified, conclusions and references added, exposition improved, typos fixed; 38 pages, 8 postscrip

Scientific paper

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 a knot in the 3-sphere its conormal bundle is perturbed to disconnect it from the zero section and then pulled through the conifold transition. The construction produces totally real submanifolds of the resolved conifold that are Lagrangian in a perturbed symplectic structure and correspond to knots in a natural and explicit way. We prove that both the resolved conifold and the knot Lagrangians in it have bounded geometry, and that the moduli spaces of holomorphic curves ending on the Lagrangians are compact in the Gromov topology.

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