Temperley-Lieb algebras are certain finite-dimensional algebras coming originally from statistical physics and knot theory. Around 2019, they became one of the first examples of homological stability for algebras (homology is here taken to be certain Tor-groups). Very little, however, is known about their homology `beyond the stable range'. I'll introduce these algebras, and give an overview of what we know about their homology, then report on some new calculations, which are joint work in progress with Rachael Boyd, Oscar Randal-Williams, and Robin Sroka.
