The minimal Rickard complexes of braids on two strands
Geometric Topology
2026-04-21 v2 Quantum Algebra
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.
Cite
@article{arxiv.2510.13764,
title = {The minimal Rickard complexes of braids on two strands},
author = {Joshua Wang},
journal= {arXiv preprint arXiv:2510.13764},
year = {2026}
}
Comments
47 pages. An explicit formula for the grading shift H is provided, simplifying and shortening sections 4.3 and 5.2