  We introduce new polynomial isotopy invariants for closed braids. They are
constructed as polynomial valued {\em Gauss diagram 1-cocycles} evaluated on
the full rotation of the closed braid $\hat \beta$ around the core of the
corresponding solid torus. They can be calculated with polynomial complexity
with respect to the braid length and their derivatives evaluated at $x=1$ are
finite type invariants of closed braids. Let the solid torus V be standardly
embedded in the 3-sphere and let L be the core of the complementary solid torus
$S^3\setminus V$. We give examples which show that a natural refinement of our
invariants can detect (even with linear complexity with respect to the braid
length if the number of strands is fixed, and with quadratic complexity if it
is not fixed) the non-invertibility of the 2-component link $\hat \beta \cup
L\hookrightarrow S^3$, what quantum invariants fail to do.
