Augmentations and rulings of Legendrian knots

For any Legendrian knot $\Lambda$ in $(\mathbb{R}^3, \mathrm{ker}(dz - ydx))$, we show that the existence of an augmentation to any field of the Chekanov–Eliashberg differential graded algebra over $\mathbb{Z}[t, t{-1}]$ is equivalent to the existence of a ruling of the front diagram, generalizing results of Fuchs, Ishkhanov, and Sabloff. We also show that any even graded augmentation must send $t$ to $-1$.

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Published 10 January 2017