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

TL;DR

This paper develops a new algebraic model for the differential graded algebra of planar loops when the dimension parameter satisfies 2n=4, incorporating natural involutions and providing a clearer description of the existing model.

Contribution

It introduces a new, more explicit model for the dga of planar loops at 2n=4, including natural involutions, advancing the understanding of its algebraic structure.

Findings

New explicit model for the dga at 2n=4

Incorporation of natural involutions into the model

Enhanced understanding of the algebraic structure of planar loops

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.