In this paper, we demonstrate that the complete, hyperbolic representation of various two-bridge knot and link groups enjoy a certain local rigidity property inside of the PGL4(R) character variety. We also prove a complementary result showing that under certain rigidity hypotheses, branched covers of amphicheiral knots admit non-trivial deformations near the complete, hyperbolic representation.

All known examples of generic 3D bar-and-joint frameworks where the distance between a non-edge pair is implied by the edges in the graph contain a rigid vertexinduced subgraph. In this paper we present a class of arbitrarily large graphs with no non-trivial vertex-induced rigid subgraphs, which have implied distances between pairs of vertices not joined by edges. As a consequence, we obtain (a...