### University of Arkansas Topology Seminar: 3/6/2014

### Speaker: Dan Rutherford

### Title: Cellular computation of Legendrian contact homology in
dimension 2

Abstract: We consider Legendrian contact homology
of a Legendrian surface, *L*, in ℝ^{5} (or more
generally in the 1−jet space of a surface). Such a Legendrian,
*L*, can be conveniently presented via its front projection which
is a surface in ℝ^{3} that is immersed except for certain
standard singularities.

Legendrian contact homology associates a
differential graded algebra (DGA) to *L*. In this setting, the
construction of the DGA was carried out by
Etnyre−Ekholm−Sullvan with the differential defined by
counting holomorphic disks in ℂ^{2} with boundary on the
Lagrangian projection of *L*. Subsequent work of Ekholm, allows
for the differential to be computed with a count of certain gradient
flow trees replacing the holomorphic disks. This simplifies matters
by replacing a PDE problem with an ODE problem. However, the required
gradient flow trees are still complicated global objects, so that
computing the differential in this manner for a given Legendrian is
far from algorithmic.

I will discuss work in progress with Mike
Sullivan. The goal is to give a computation of the DGA of *L* by
starting with a cellular decomposition of the base projection (to
ℝ^{2}) of *L* that contains the projection of the
singular set of *L* in its 1−skeleton. Generators are
associated to each cell, and the differential is determined in a
formulaic manner by the nature of the singular set above the boundary
of a cell.

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