Classification of topological trivial matter with non-trivial defects.

In this paper, we apply the K-theory to classify topological trivial fermionic phases which, nonetheless, host symmetry-protected non-trivial defects. An important implication of our work is that the existence of Majorana zero mode in the vortex core is neither a necessary nor a sufficient condition for the superconductor in question being topologically non-trivial.