UC Berkeley

The central motivation of this dissertation is a questionof Gordon asking if an infinite descending chain of ribbon concordances $K_0 \geq K_1 \geq \ldots$ is eventually constant. Restricting to the case when these knots are hyperbolic, we approach this problem by studying relations between the $\SL_2 \C$ representation varieties of ribbon concordant knots. We first rule out one potential approach by giving examples of ribbon concordances that do not induce surjections on representation varieties or character varieties. Then we provide two sufficient conditions on the $K_i$ for Gordon's question to have a positive answer. The first condition is if the $K_i$ satisfy a conjecture of Chinburg, Reid, and Stover. Using the theory of deformations of cone manifolds, we show prove this conjecture in the case that a knot admits a Euclidean cone structure with cone angle $\alpha \leq \pi$. The second condition is when a faithful representation and a reducible representation lie on the same component of the chararcter variety. This result is shown by making use of homology with local coefficients. In particular, these conditions show that any descending chain of 2-bridge hyperbolic knots is eventually constant.