Mutation is an operation on the set of 3-manifolds that cut a 3-manifold along a surface of genus $\leq 2$ and reglue using the hyperelliptic involution. Two 3-manifolds are said to be mutative if they are related by a finite sequence of mutations. Mutative 3-manifolds share many topological invariants such as Gromov's norm, and $SU(2)$-quantum invariants.

In this talk we study the equivalence relation of being mutative. Various properties in terms of geometric structures will be discussed. If $M$ has one of Thurston's eight geometries, then any manifold mutative to $M$ has the same geometry as that of $M$. If $M$ has the elliptic geometry, then $M$ is the only manifold mutative to $M$. If $M$ is a Seifert fibered space, then $M$ is mutative to at most three different 3-manifolds; and if the genus of the base orbifold is positive, then $M$ is mutative to at least one other manifold. For hyperbolic 3-manifolds, we show that there are examples of arbitrarily many manifolds that are mutative to each other. The same is true for other classes of Haken 3-manifolds such as the graph manifolds.

As a consequence, we get many examples of 3-manifolds with the same $SU(2)$-quantum invariants, as well as many examples of hyperbolic 3-manifolds with the same volume.