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The dga of planar loops when $2n=4$

Guy Boyde

TL;DR

This work analyzes the differential graded algebra of planar loops $L(2n)$ and its two reflection involutions, focusing on the case $2n=4$. It provides an explicit model $(A,d,\sigma_{\updownarrow},\sigma_{\leftrightarrow})$ with generators $(x,\hat{x},r,y)$ and a concrete differential, together with a weak equivalence to the existing $L(4;R,a)$ model via $\varphi$, and a comparison map $\psi$ linking to the classic Tempeley–Lieb framework. A reduced bar construction yields a combinatorial basis of graffiti elements, which are organized via a filtration by dividers and analyzed through a divider-based spectral sequence; the main technical work computes homology for low-loop cases and establishes vanishing results for higher loops. The analysis uses a word-based re-encoding of diagrams (letters and pivots) and shows that the $E^2$ page is a tensor algebra on key classes, aligning the new model with the known one. Overall, the paper clarifies the action of the involutions on homology, provides an explicit $2n=4$ model, and sets a template for deeper comparisons in the Temperley–Lieb setting with practical calculational benefits.

Abstract

The dga of planar loops was introduced in recent work of the author, Boyd, Randal-Williams, and Sroka, where a minimal model for it was given. This dga enjoys two natural `reflection' involutions. In the first nontrivial case, $2n=4$, we give a new model which incorporates these involutions, as well as a more explicit description of the existing model.

The dga of planar loops when $2n=4$

TL;DR

This work analyzes the differential graded algebra of planar loops and its two reflection involutions, focusing on the case . It provides an explicit model with generators and a concrete differential, together with a weak equivalence to the existing model via , and a comparison map linking to the classic Tempeley–Lieb framework. A reduced bar construction yields a combinatorial basis of graffiti elements, which are organized via a filtration by dividers and analyzed through a divider-based spectral sequence; the main technical work computes homology for low-loop cases and establishes vanishing results for higher loops. The analysis uses a word-based re-encoding of diagrams (letters and pivots) and shows that the page is a tensor algebra on key classes, aligning the new model with the known one. Overall, the paper clarifies the action of the involutions on homology, provides an explicit model, and sets a template for deeper comparisons in the Temperley–Lieb setting with practical calculational benefits.

Abstract

The dga of planar loops was introduced in recent work of the author, Boyd, Randal-Williams, and Sroka, where a minimal model for it was given. This dga enjoys two natural `reflection' involutions. In the first nontrivial case, , we give a new model which incorporates these involutions, as well as a more explicit description of the existing model.

Paper Structure

This paper contains 17 sections, 28 theorems, 35 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Temperley--Lieb diagrams, cell modules, ideals
  3. Cell modules
  4. Ideals
  5. The reduced complex of loops
  6. The juxtaposition product, and counting loops
  7. The filtration by dividers
  8. The spectral sequence
  9. Proof of \ref{['thm:mainTechnical']}
  10. A basic observation
  11. A change of perspective
  12. Part (\ref{['part:1']})
  13. Part (\ref{['part:2']})
  14. The nested summand
  15. The unnested summand
  16. ...and 2 more sections

Key Result

Theorem 1.1

The map of dgas defined on generators by (commutes with both involutions and) is a weak equivalence.

Figures (3)

  • Figure 1: $\mathrm{(a)}$ the action of the differential on a system of loops in $L(4)_3$, and $\mathrm{(b)}$ The juxtaposition product.
  • Figure 2: The page $E^1_{p,q}$ of the spectral sequence computing the homology of $C_*[2,0]^{\mathrm{in}}$.
  • Figure 3: The page $E^1_{p,q}$ of the spectral sequence computing the homology of $C_*[2,0]^{\mathrm{out}}$.

Theorems & Definitions (57)

  • Theorem 1.1
  • Proposition 1.2
  • Lemma 2.1
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Remark 3.4
  • Remark 4.1
  • Lemma 5.1
  • proof
  • ...and 47 more