The classification of links up to clasp-pass moves

We give a complete classification of links up to clasp-pass moves, which coincides with Habiro’s $C_3$-equivalence. We also classify links up to band-pass and band-# moves, which are versions of the usual pass- and #-move, respectively, where each pair of parallel strands belong to the same component. This recovers and generalizes widely a number of partial results in the study of these local moves. The proofs make use of clasper theory.