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In two former papers, the authors independently proved that the space ofhyperbolic cone-3-manifolds with cone angles less than 2{\pi} and fixedsingular locus is locally parametrized by the cone angles. In this sequel, weinvestigate the local shape of the deformation space when the singular locus isno longer fixed, i.e. when the singular vertices can be split. We show that thedifferent possible splittings correspond to specific pair-of-pantsdecompositions of the smooth parts of the links of the singular vertices, andthat under suitable assumptions the corresponding subspace of deformations isparametrized by the cone angles of the original edges and the lengths of thenew ones.

Complex twist flows on surface group representations and the local shape of the deformation space of hyperbolic cone-3-manifolds