Constructing a free resolution and remarks on bar homology of Temperley-Lieb algebra $TL_3$

Abstract

In this paper I try to construct the Anick's resolution for the Temperley-Lieb algebra $TL_3$ and bar homology $Tor_\ast^{TL_3}(\mathbb{C},\mathbb{C})$ and shows that how $\tau \in \mathbb{C}$ given in the definition of $TL_3$ is actively involved in the differentials both in free resolution and bar homology. It suggests that the bar homology depends on the nature of $\tau$.