OK, unless I have miscalculated, the Jones polynomial for the knot you 
drew obeys

V(t)/t = t + t^(1/2) - t^(-1/2)

which, unless I'm very much mistaken, rearranges to

V(t) = t^2 + t^(3/2) - t^(1/2)

For comparism, the polynomial for the right-trefoil is

V(t) = -t^4 + t^3 + t

For the left-trefoil it is

V(t) = -t^(-4) + t^(-3) + t^(-1)

Most importantly, the Jones polynomial for the unknot is V(t) = 1. The 
Jones polynomial for your knot is NOT 1 - therefore your knot is 
non-trivial. QED.

Andrew.

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