Bending Representations
Here's a very well-studied construction that I think everyone should know, called bending representations . I'll start with the algebraic picture, and then get into the relationship with topology. (At the end we'll hopefully prove that a splice of two knots in the three-sphere has positive dimensional $SU(2)$ character variety.) First, recall that given a group $G$, an $SU(2)$ representation is just a homomorphism $\rho: G \to SU(2)$. The representation variety is $R(G) = Hom(G,SU(2))$. Now $SU(2)$ acts on $R(G)$ by conjugation: for $A \in SU(2)$, define $\rho^A(g) = A \rho(g) A^{-1}$. We define the $SU(2)$ character variety of $G$, $\chi(G)$, to be the quotient of $R(G)$ by conjugation. Note that $\rho^{\pm Id} = \rho$. A representation $\rho$ is irreducible if $\rho^A = \rho$ implies $A = \pm I$. For some motivation to those topologically inclined, one of the Kirby problems is to prove that every homology three-sphere other...