I’ve posted code for computing Legendrian knot invariants in Sage on my website: https://web.math.princeton.edu/~ssivek/code/lch.sage This is an update of code I had previously written and posted in the same location, with a bunch of new material for computing the augmentation categories of NRSSZ. It includes the table used by Melvin-Shrestha (also available in the “Legendrian …

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Let’s assume for now that if $\Lambda$ is a Legendrian knot in the standard $\mathbb{R}^3$, then the groupoid cardinality of $Aug(\Lambda,\mathbb{F}_q)$ is given by $$|Aug(\Lambda,\mathbb{F}_q)| = R_\Lambda(q^{1/2}-q^{-1/2}) \cdot \frac{q^{(tb(\Lambda)+1)/2}}{q-1}.$$ My convention for the ruling polynomial $R_\Lambda(z)$ is that a ruling with s switches and c cusps contributes $z^{s-c+1}$, so for example the $tb=-1$ unknot $U$ …

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