Talk:Temperley–Lieb algebra

I'm hoping someone who is more familiar with formatting and writing wikipedia pages will clean this up, but I thought adding the images will make it easier to write about the graphical representation. I might have a few more kicking around if there's something that someone wants, in particular the half-diagrams for the representation theory. I'm also not sure if the examples are appropriate, but they're the best way to make the relations clear. Magnus the Great (talk) 16:08, 29 July 2010 (UTC)[reply]

Error in definition of Temperley-Lieb Hamiltonian

The definition of $e_{j}$ in the section on the Temperley-Lieb Hamiltonian is clearly wrong, since it should depend on $j$ . I couldn't find anything like this in the cited reference, either. In what follows it looks like they take $e_{j}=U_{j}$ which seems reasonable. If someone can provide a reference which explicitly gives some Hamiltonian, that would be ideal. Ctourneur (talk) 16:27, 14 October 2014 (UTC)[reply]

Indeed. Fixed $e_{j}$ to $U_{j}$ and added $\delta =1$ . In general the parameter $\delta$ is arbitrary, but for the combinatorial applications that for some reason this page focuses entirely on, one needs $\delta =1$ . 128.250.0.101 (talk) 04:24, 30 September 2019 (UTC)[reply]

Jones polynomial

It would be nice to connect this with the Jones polynomial. 2602:24A:DE47:BB20:50DE:F402:42A6:A17D (talk) 10:52, 14 September 2020 (UTC)[reply]

I've run across several references to this connection, but I don't yet feel like I can write the connection down.

Kaufmann’s Knots and Physics might be a good source here. He connects Jones and Temperley-Lieb-Lieb on page 100, and elsewhere in the book he covers each at length and in language that seems to me to be compatible.

Tangentially, he also covers (p.605) the connection of TL algebras to parenthesis structures, which seems like another valuable connection to include, given the use of such things in modern computer programming and markup processing. (Are they also what Ridout and Saint-Aubin (2014) refer to as “link states”?)

— Octavo (talk) 17:06, 7 March 2022 (UTC)[reply]

Nice suggestions. I have seen link states written with parentheses so they are probably related to parenthesis structures. If you cannot write the technical statements, could you at least write a general statement about the existence of a connection, and cite your sources? Sylvain Ribault (talk) 20:34, 7 March 2022 (UTC)[reply]