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Regarding Rick T's comment:often
Perhaps one of the confusions here (at least for me) is that looking at
Bruce's link (below) it is now clear that the claim is NOT that the
gymnast/diver imparts NO angular momentum on takeoff, only that they
impartimpart no angular momentum around the vertical axis--that is, they do
.......AM around a horizontal axis perpendicular to their direction of motion.
Total angular momentum is a (bi)vector. As such all of its components
are *individually conserved* in the absence of corresponding components
of the externally applied net torque.