### University of Arkansas Topology Seminar: 9/26/2013

### Speaker: Dan Rutherford

### Title: Rulings polynomials and augmentations into finite
fields

Abstract: A normal ruling is a type of
decomposition of the front diagram of a Legendrian knot in
ℝ^{3}. Making a refined count of normal rulings allows
for the definition of a Legendrian link invariant known as the ruling
polynomial that is a Laurent polynomial in a variable, *z*.
Another class of Legendrian invariants, the augmentation numbers,
arise from making normalized counts of augmentations of the Legendrian
contact homology DGA into finite fields. We show that augmentation
numbers are determined by specializing the ruling polynomial via
*z* =
*q*^{½}−*q*^{−½}
where *q* denotes the order of the corresponding finite field.
As a corollary, we deduce that the ruling polynomial is determined by
the Legendrian contact homology DGA. This is joint work with Brad
Henry.

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