Algebraic Weaves and Braid Varieties

Abstract

In this manuscript we study braid varieties, a class of affine algebraic varieties associated to positive braids. Several geometric constructions are presented, including certain torus actions on braid varieties and holomorphic symplectic structures on their respective quotients. We also develop a diagrammatic calculus for correspondences between braid varieties and use these correspondences to obtain interesting stratifications of braid varieties and their quotients. It is shown that the maximal charts of these stratifications are exponential Darboux charts for the holomorphic symplectic structures, and we relate these strata to exact Lagrangian fillings of Legendrian links.

Lagrangian Fillings in A-type and their Kalman Loop Orbits

“…This characterization leads to the appearance of a key supporting character, Euler's identity for continuants. Continuants naturally appear in the definition of the augmentation variety of λ(A n−1 ) [CGGS20], and in Section 3 we show that the action of the Kalman loop is identical to Euler's continuant identity. In this sense, we may interpret the Kalman loop action on the augmentation variety as a Floer-theoretic manifestation of Euler's identity for continuants.…”

“…In order to prove Theorem 1.1 we will require a convention for relating trivalent vertices of a 2-graph to crossings in the Ng resolution. The vertical weave construction given originally in [CGGS20] allows us to unambiguously associate a trivalent vertex in the weave to the 0-resolution of a specific crossing in a pinching sequence filling by specifying a convention for breaking the symmetry of a 2-graph inscribed in a triangulated (n + 2)-gon.…”

“…See Figure 4 for an example in the case n = 6. Note that this choice differs from the convention in [CGGS20], as the choice of labeling given there corresponds to resolving the leftmost crossing of the pair in the D − 4 cobordism.…”

“…For λ(A n−1 ), the polynomials defining the augmentation variety have a combinatorial description as a specific entry in a product of matrices. These matrices originally appeared in [Kál06] and were used in [CGGS20] to prove a holomorphic symplectic structure on the augmentation variety. We adopt the convention of [CGGS20] and define the braid matrix B(z i ) = 0 1 1 z i .…”

“…These matrices originally appeared in [Kál06] and were used in [CGGS20] to prove a holomorphic symplectic structure on the augmentation variety. We adopt the convention of [CGGS20] and define the braid matrix B(z i ) = 0 1 1 z i . The augmentation variety of the max-tb Legendrian (2, n) torus link is then cut out as the zero set of the polynomial…”