The homology of a Temperley-Lieb algebra on an odd number of strands (Q6632138)

In this paper, the author shows that the homology of any Temperley-Lieb algebra on an odd number of strands vanishes in positive degrees. The first section is devoted to the notion of the Temperley-Lieb algebra \(\mathcal{T}\mathcal{L}n(a)\) and of its trivial module. Then, the author introduces an important class of left submodules of \(\mathcal{T}\mathcal{L}n(a)\). These modules are used to obtain the cellular Davis complex. The homology of \(\mathcal{T}\mathcal{L}n(a)\) is then studied. This paper improves a result obtained by \textit{R. Boyd} and \textit{R. Hepworth} [Geom. Topol. 28, No. 3, 1437--1499 (2024; Zbl 1540.20103)]. Alternative for two of their results are also given.