Full exceptional collections on Fano threefolds and groups generated by spherical twists on K3 surfaces

For a Fano threefold admitting a full exceptional collection of vector bundles of length four we show that all full exceptional collections consist of shifted vector bundles. We prove this via a detailed study of the group generated by spherical twists on an anticanonical divisor. For example, we prove that this group is free and provide explicit generators.