# Jones Polynomial of the trace closure of the fundamental braid

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.

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If you were to follow the nth Jones-wenzl idempotent by that braid, close it up and take the bracket, you could compute that easily. –  Charlie Frohman Aug 3 '11 at 2:28
For $n$ even, it is a torus link of $n/2$ components, and for $n$ odd it is a cable link. You can convince yourself easily that the braid closure may be drawn on a standard Mobius strip, which sits nicely inside a solid torus with boundary a $(1,2)$ curve on the boundary torus. When $n$ is even, the strands are parallel, and may be pushed onto parallel curves on the boundary torus. When $n$ is odd, one may do this for all but one strand, which is isotopic to the core of the Mobius strip, and therefore it is cabled. –  Ian Agol Aug 30 '11 at 14:40