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

TL;DR

This paper constructs Anick's free resolution and examines bar homology of the Temperley-Lieb algebra TL_3, revealing how the parameter tau influences the algebra's homological properties.

Contribution

It provides an explicit construction of the free resolution and analyzes the dependence of bar homology on the parameter tau in TL_3.

Findings

Bar homology depends on tau's nature

Explicit Anick's resolution for TL_3 constructed

Tau influences differentials in homology computations

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 $\tau$.