Character varieties of knot complements and branched double-covers via the cord ring

We study the relationship between Ng's abelian cord ring and SL(2,C) characters of the two-fold branched cover $\Sigma(K)$. Augmentations, and their corresponding rank, play a central role in the relationship. Our study also leads to a correspondence between trace-free SL(2,C) characters of a knot complement and augmentations. Whether the trace-free character is metabelian or not is determined by the rank of the corresponding augmentation. Necessary and sufficient conditions are given on the augmentation for it to correspond to an SU(2) character. This provides an effective tool for finding representations of relevance to singular instanton homology. We also give necessary and sufficient conditions for an augmentation to induce a non-abelian SL(2,R) character of $\Sigma(K)$ and exploit this to discuss branched covers with left-orderable fundamental group.