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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 theircorresponding rank, play a central role in the relationship. Our study alsoleads to a correspondence between trace-free SL(2,C) characters of a knotcomplement and augmentations. Whether the trace-free character is metabelian ornot is determined by the rank of the corresponding augmentation. Necessary andsufficient conditions are given on the augmentation for it to correspond to anSU(2) character. This provides an effective tool for finding representations ofrelevance to singular instanton homology. We also give necessary and sufficientconditions for an augmentation to induce a non-abelian SL(2,R) character of$\Sigma(K)$ and exploit this to discuss branched covers with left-orderablefundamental group.

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