The group generated by spherical twists on a Picard-rank-one K3 surface is free, and this proves transitivity and freeness of braid mutations on full exceptional collections for the four Fano threefolds with such a collection of four vector bundles.
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Groups generated by spherical twists on K3 surfaces and full exceptional collections on Fano threefolds
The group generated by spherical twists on a Picard-rank-one K3 surface is free, and this proves transitivity and freeness of braid mutations on full exceptional collections for the four Fano threefolds with such a collection of four vector bundles.