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The minimal Rickard complexes of braids on two strands

Joshua Wang

TL;DR

The article constructs explicit minimal complexes for two-strand braids colored by positive integers, organizing them into a family ${}^b_a\mathscr{P}^c_d$ with a filtration whose initial pieces replicate Rickard complexes. A key auxiliary complex $\mathscr{K}$ is built from webs/foams and then extended to $\mathscr{P}$, yielding a bounded-above, minimal complex whose leftmost truncations correspond to Rickard-type braids; for the common case of a=b=c=d, the construction yields an idempotent projector onto the two-column highest-weight representation. The main contributions are the explicit, symmetry-rich formulas for the differentials, the perverse-filtration-based minimality proof, and the demonstration that these complexes realize colored $\mathfrak{sl}_N$ homology computations via braid closures. Collectively, the results provide a computationally effective, structurally transparent framework for understanding colored triply-graded homology on two-strand braids and their closures.

Abstract

The Rickard complex of a braid with strands colored by positive integers is a chain complex of singular Soergel bimodules. The complex determines the colored triply-graded homology and colored sl(N) homology of the braid closure, when closure is color-compatible. For each braid on two strands with any colors, we construct a minimal complex that is homotopy equivalent to its Rickard complex. It is not obtained by laborious simplification; instead, it is defined directly by explicit formulas obtained by educated guesswork and reverse engineering.

The minimal Rickard complexes of braids on two strands

TL;DR

The article constructs explicit minimal complexes for two-strand braids colored by positive integers, organizing them into a family with a filtration whose initial pieces replicate Rickard complexes. A key auxiliary complex is built from webs/foams and then extended to , yielding a bounded-above, minimal complex whose leftmost truncations correspond to Rickard-type braids; for the common case of a=b=c=d, the construction yields an idempotent projector onto the two-column highest-weight representation. The main contributions are the explicit, symmetry-rich formulas for the differentials, the perverse-filtration-based minimality proof, and the demonstration that these complexes realize colored homology computations via braid closures. Collectively, the results provide a computationally effective, structurally transparent framework for understanding colored triply-graded homology on two-strand braids and their closures.

Abstract

The Rickard complex of a braid with strands colored by positive integers is a chain complex of singular Soergel bimodules. The complex determines the colored triply-graded homology and colored sl(N) homology of the braid closure, when closure is color-compatible. For each braid on two strands with any colors, we construct a minimal complex that is homotopy equivalent to its Rickard complex. It is not obtained by laborious simplification; instead, it is defined directly by explicit formulas obtained by educated guesswork and reverse engineering.
Paper Structure (21 sections, 25 theorems, 171 equations, 9 figures)

This paper contains 21 sections, 25 theorems, 171 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symmetric polynomials in differences of alphabets
  4. The nil-Hecke algebra
  5. Singular Bott--Samelson bimodules
  6. Maps between singular Bott--Samelson bimodules
  7. Construction of $\mathscr{K}$
  8. The webs and foams in $\mathscr{K}$
  9. The shape of $\mathscr{K}$
  10. The objects of $\mathscr{K}$
  11. The differential of $\mathscr{K}$
  12. Construction of $\mathscr{P}$
  13. The webs and foams in $\mathscr{P}$
  14. The shape of $\mathscr{P}$
  15. The objects of $\mathscr{P}$
  16. ...and 6 more sections

Key Result

Theorem 1.1

The complex $\mathscr{P}_b$ has the following properties.

Figures (9)

  • Figure 1: A web. Edges are always oriented from right to left.
  • Figure 2: Alphabets assigned to the edges of the web of Figure \ref{['fig:exampleWeb']}.
  • Figure 3: The singular Bott--Samelson variety assigned to the web of Figure \ref{['fig:exampleWeb']}.
  • Figure 4: The coarse and fine cubulations of $[0,3]^b \subset \mathbf{R}^b$ for $b = 2$.
  • Figure 5: On the left are the vertices $\varepsilon \in [0,3]^b \cap \mathbf{Z}^b$ for $b = 2$. In the middle are the values of $r(\varepsilon) \in \{0,1,2\}$. On the right are the values of $G(\varepsilon) \in \mathbf{Z}$ for $a = b = c = d = 2$.
  • ...and 4 more figures

Theorems & Definitions (78)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Example 1.4
  • Example 1.5
  • Remark 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • ...and 68 more