(trigger warning: M-theory.) I just learned what a “brane tiling” is, first from a talk of Junya Yagi about this paper, and then some more from this nice survey of Masahiko Yamazaki. (My attempt to tell) the story is the following. You make a certain configuration of M5 branes in M theory, of the following sort: …
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 …
In [STZ], we observed a connection between the sheaf category on an n-strand braid with identically zero Maslov potential, and certain constructions related to the geometric representation theory of gl(n). Specifically, the relationship was the following. First, consider the $n$-line category, i.e., the braid in question contains no crossings at all, carrying Maslov potential $\mu = …
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 $U$ …
At the recent AMS sectional meeting in Las Vegas, Steven and Lenny gave talks about the augmentation category, and why it’s isomorphic to the sheaf category [NRSSZ]. Here are their slides: Lenny’s talk: The augmentation category of a Legendrian knot Steven’s talk: Augmentations are Sheaves (I also gave a talk, but about moduli of objects in these categories …
Finally, let’s prove this counting formula. Theorem. $$\# \underline{Aug}(\mathbb{F}_q) = q^{(tb-\chi^*)/2 } \cdot (q – 1)^{-c} \cdot \# Aug(\mathbb{F}_q)$$ Proof. Note that the Thurston-Bennequin number is also $tb = \sum_{i} (-1)^{i} a_i$, so that $$(tb-\chi^*)/2 = \sum_{i < 0} (-1)^{i} a_i$$ Recall that the complex we use to compute $Aug_+$ hom spaces has generators in degrees shifted up one …
I’m going to try and fill in the fudge factors. There is a unique way of doing this, given known results and assuming the only allowed constants are the Thurston-Bennequin number and the number of components. First let me assemble the known results. We take out of [HR]: Definition. For a ruling $\rho$, we write $-\chi(\rho)$ for the …
Recall how Deligne associates a filtered sheaf to an irregular connection: along each direction coming into the singularity, you can filter the space of local analytic solutions by growth rate. (Maybe it would be better to say something about filtering the differential Galois group, in terms of what comes next). This filtration of a sheaf …
A Morse complex over $\mathbb{Z}$ is like a sheaf on a Legendrian front on $Spec(\mathbb{Z}) \times \mathbb{R}$ which is coherent along $Spec(\mathbb{Z})$ and constructible along $\mathbb{R}$. Is that a hint for classifying such Morse complexes? Any such Morse complex determines a (finite) collection of primes: those at which the ruling is over $\mathbb{F}_p$ is not …
Lemma. Fix $\Lambda \subset T^{\infty, -} R$ for $\Lambda$ a collection of points at minus infinity. Fix a grading on these points, let $\mu$ be the corresponding Maslov potential. Split these into some pairs $(a_i, b_i)$ such that $a_i < b_i$ and some singletons $c_j$. Then $$\bigoplus_i k_{(a_i, b_i]}[m_i] \oplus \bigoplus_j k_{(c_j, \infty]}[n_j] \in Sh_{\Lambda, \mu}(R, k)_0$$ The …
