The fundamental braid $\Delta_n \in B_n$ is simply a twist by $\pi$ applied to the entire row of $n$ strands. In terms of Artin generators, it is given by $$ \Delta_n = (\sigma_1 \sigma_2 \cdots \sigma_{n-1})(\sigma_1 \sigma_2 \cdots \sigma_{n-2})\cdots (\sigma_1 \sigma_2) \sigma_1~. $$ The square of $\Delta_n$ (i.e., the full $2\pi$ twist) generates the center of $B_n$.

I have a rather simple (and quite possibly trivial) question about these braids. What is the Jones polynomial of the trace closure of $\Delta_n$? Do the trace closures of the $\Delta_n$ result in some well-known link family?

I have tried computing the J.P. in the obvious way using the Kauffman bracket; some simplifications are possible, but so far nothing sufficient to lead to a general formula.