Consider a knot in a 3-manifold. Witten tells us to make invariants of this by considering some big integral over connections, with an insertion giving the holonomy around the knot. Usually you are allowed to color this holonomy on the knot by a representation. Apparently also you can ask the connection to be singular along …

# Manifolds with Legendrian isotopic boundaries are h-cobordisms

Related to the previous post, I have been thinking about when, given submanifolds $A, B \subset M$, one can have a Legendrian isotopy from $S^*_A M$ to $S^*_B M$. The simplest kind of such isotopy is one which begins and ends with Reeb flow, and in the interim, remains in the complement of both. For …

# A funny isotopy on the 3-sphere

In the papers on the conormal torus (I, II), we showed that if two knots $K, K’ \subset \mathbb{R}^3$ have Legendrian isotopic conormal tori $T, T’$ then the knots themselves are isotopic. Usually the difference between $\mathbb{R}^3$ and $S^3$ is not important in knot theory, but it’s not so clear if that’s the case here: …

# The conormal torus is a complete knot invariant

In case you missed the arxiv posting…

# Cohomology of the big positroid stratum

Theorem. Assume $k, n$ are coprime. Let $X_{k,n}$ denote the quotient of the big positroid cell of $Gr(k, k+n)$ by the appropriate torus action; i.e., “the image of the X variety in the A variety”. Then the cohomology of $X_{k,n}$ is all in even degrees, and the $2i$’th Betti number are given by counting the …

# Complete cluster structures from Legendrian knots

In some recent papers, we considered moduli spaces $M(\Sigma, \Lambda)$ (I) or more generally $M(\mathbb{L})$ (II) associated to certain configurations in 4d symplectic geometry. Basically the point was that in this setting, finding an exact Lagrangian in the geometry gives rise to an algebraic torus chart, and disk surgery gives rise to cluster transformations of charts. Here …

# Immersed Lagrangians from Skeleta

Some augmentations of the Legendrian DGA come from smooth fillings; in “augmentations are sheaves” we showed that, at least for Legendrian knots in the standard contact 3-space, all augmentations are geometric in the sense that the category of augmentations is equivalent to a geometrically defined category of sheaves. I will outline a strategy for making this into the perhaps more …

# On Cornwell’s constructions

I want to record here some ideas on interpretations of the work of Chris Cornwell; these come from discussions with him and others. Mostly I am concerned with the paper where he constructs representations of $\pi_1(\mathbb{R}^3 \setminus K)$ from augmentations of the Legendrian contact algebra $A(S^*\mathbb{R}^3; S^*_K \mathbb{R}^3)$ of the conormal torus. This gives an inverse to a …

# The support of an augmentation

Let $\Lambda \subset J^1(X)$ be Legendrian. In an earlier post, I gave a conceptual argument for why there should be a fully faithful morphism from the category $Aug(\Lambda)$ defined as in [NRSSZ] to the category $Shv_\Lambda(X \times \mathbb{R})$ of sheaves on $X \times \mathbb{R}$ with microsupport at infinity contained in $\Lambda$. Here I will explain that …

# Morse realization of skeletal moves: baby case.

This post is a toy investigation of the cluster move as a transformation of skeleta. It contains any new ideas. The cluster move should represent the relationship between opposite ends of a continuous process labeled by a one-parameter family of Morse functions whose skeleta transform by a blow-down, blow-up procedure. Arborealization is realized by the blow-up, so …