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On the Cohomology of Two Stranded Braid Varieties

Tonie Scroggin

Abstract

We compute the cohomologies of two strand braid varieties using the two-form present in cluster structures. We confirm these results with proof using Alexander and Poincaré duality. Further, we consider products of braid varieties and their interactions with the cohomologies.

On the Cohomology of Two Stranded Braid Varieties

Abstract

We compute the cohomologies of two strand braid varieties using the two-form present in cluster structures. We confirm these results with proof using Alexander and Poincaré duality. Further, we consider products of braid varieties and their interactions with the cohomologies.
Paper Structure (13 sections, 36 theorems, 191 equations, 9 figures)

This paper contains 13 sections, 36 theorems, 191 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Braid varieties
  3. Definition of braid varieties
  4. Cohomology using Alexander and Poincaré duality
  5. The Grassmannian and the open positroid variety
  6. Cluster algebras
  7. Ring structure on cohomology using (algebraic) deRham cohomology
  8. Constructing the forms
  9. de Rham cohomology
  10. Examples
  11. Performing cuts
  12. Cuts for braid varieties
  13. Cuts, forms and cohomology

Key Result

Theorem 1.1

The braid variety $X(\sigma^k)$ is defined in $\mathbb{C}^k$ by the equation $F_k(z_1,\ldots,z_k)=0$ where $F_k$ is given by the recursion where with initial values $F_1(z_i)=z_i$, $F_0\equiv1$ and $F_{-1}\equiv0$. Moreover, if $F_k(z_1,\dots,z_k)=0$, then $F_{k-1}(z_1,\dots,z_{k-1})\neq0$ and

Figures (9)

  • Figure 1: Examples of two diagonal cuts. The top is shows a Type A cut and the bottom shows a Type B cut.
  • Figure 2: The braid $\sigma_1^4$ with each crossing $j$ labeled with a complex variable $z_j$.
  • Figure 3: Section of the triangulation of $\mathrm{U_{fan}}$, see Figure \ref{['fig:Ufan']}, between the vertices $1,k-1,k$ and $k+1$
  • Figure 4: The special chart $\mathrm{U_{fan}}\in\Pi_{2,k+1}^{\circ,1}$ where each of the $k-2$ diagonals are have fixed endpoint at $v_1$. The Pluc̈ker coordinates, or cluster variables, correspond to the weights of the edges given by either a blue square (frozen vertices) or a green circle (mutable vertices). The quiver of the cluster chart is generated by clockwise orientation of the colored arrows in each triangle of the triangulation. This procedure produces the quiver $A_{k-1}$, seen in purple, with $w$ as the singular frozen variable. In the terminology of CGGLSS, this chart is given by the right inductive weave.
  • Figure 5: The two cluster charts for the braid variety $X(\sigma^3)$. On the left is chart $U_1$ where the vectors $v_i\in\Pi_{2,4}^{\circ,1}$ for $1\le i\le 4$ correspond to the vertices of the polygon. The purple arrow depicts the Dynkin diagram $A_1$ with a frozen. On the right is chart $U_2$ which corresponds to the mutation of chart $U_1$.
  • ...and 4 more figures

Theorems & Definitions (72)

  • Theorem 1.1: HughesH, Chantraine-Ng-SivekCNS
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4: TrinhT
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 62 more