TITLE: What is the image of the half/full twist in the Hecke algebra, in the Kazhdan-Lusztig basis?  What is the corresponding complex of Soergel bimodules?
QUESTION [6 upvotes]: Let $B_n$ be the braid group on $n$ strands.  It has generators $\tau_i$ for $i = 1,\ldots,n-1$ which exchange the $i$th and $(i+1)$st strands, and which satisfy the relations
$\tau_i \tau_j = \tau_j \tau_i$ for $|i-j| \ne 1$ and $\tau_i \tau_{i+1} \tau_i = \tau_{i+1} \tau_i \tau_{i+1}$.  
$B_n$ maps to the Hecke algebra $H_n$, which has the same generators and the additional relation $(\tau_i-1)(\tau_i + q) = 0$.  Composing this map with the Jones-Ocneanu trace yields the HOMFLY polynomial of the closure of a given braid. 
In the braid group, there is the half-twist $\Delta$.  It exchanges the $i$-th and $(n-i)$th strands by rotating the whole braid counterclockwise.  Its square, the full twist, generates the center of the braid group.
One particularly natural choice of basis for the Hecke algebra is the Kazhdan-Lusztig canonical basis.  


What is the image of $\Delta^k$ in $H_n$, in the Kazhdan-Lusztig basis?


This is a decategorified version of the question I am really interested in:


What is the complex of Soergel bimodules corresponding to $\Delta^k$ ?


A related question, in case anyone happens to know: 


What is the HOMFLY homology of a torus knot?

REPLY [4 votes]: As Geordie says, this is really hard, and will get much harder as n increases (I'd guess $n=3$ is about an order of magnitude harder than $n=2$, and that trend continues). 
The relevant papers on this are Stosic:Homological thickness and stability of torus knots and Rozansky:An infinite torus braid yields a categorified Jones-Wenzl projector.  Jake Rasmussen had some similar results, but it looks like he never wrote them up (I guess fatherhood has been distracting for him).  In fact, you might try emailing Marko; it's possible that the thick/Schur categorification story can actually help here and make things a bit more concrete.  I wouldn't hold your breath, though.
Perhaps the most famous fact about $\Delta^k$ is that for the correct power of $q$, the limit $\lim_{k\to\infty}q^?\Delta^k$ is actually the Jones-Wenzl projector.  Lev's paper is about how this holds in the categorified picture as well, though he's working in the Temperley-Lieb case, not full Soergel bimodules.