On Newton polytopes of Legendrian knots.

Orsola Capovilla Searle (UC Davis)


Legendrian submanifolds are an important object of study in contact geometry. Wavefronts are an example of a Legendrian submanifold. A Legendrian knot in $(\mathbb{R}^3, \xi=ker(dz-ydx))$ is a smooth knot whose tangent vectors must lie in the contact hyperplanes $\xi$. As with smooth knots, Legendrian knots can bound surfaces in the four-ball. Such a surface is called a filling of the knot. For Legendrian knots we require the surfaces to be exact and Lagrangian. An important problem in contact topology is the classification of exact Lagrangian fillings of Legendrian knots in the standard contact 3-sphere up to Hamiltonian isotopy. In joint work with Roger Casals we show that Newton polytopes can be used to distinguish infinitely many exact Lagrangian fillings of Legendrian links in the standard contact $3$-sphere and higher dimensional Legendrian spheres in the standard contact $(2n+1)$ sphere up to Hamiltonian isotopy. We also show that there exist Legendrian links with infinitely many exact non-orientable Lagrangian fillings.

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