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?