On the canonical components of character varieties of hyperbolic 2-bridge link complements

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2011-08

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This dissertation concerns the study of canonical components of the SL(2, C) character varieties of hyperbolic 3-manifolds. Although character varieties have proven to be a useful tool in studying hyperbolic 3-manifolds, very little is known about their structure. Chapter 1 provides background on this subject. Chapter 2 is dedicated to the canonical component of the Whitehead link. We provide a projective model and show that this model is isomorphic to P^2 blown up at 10 points. The Whitehead link can be realized as 1/1 Dehn surgery on one cusp of both the Borromean rings and the 3-chain link. In Chapter 3 we examine the canonical components for the two families of hyperbolic link complements obtained by 1/n Dehn filling on one component of both the Borromean rings and the 3-chain link. These examples extend the work of Macasieb, Petersen and van Luijk who have studied the character varieties associated to the twist knot complements. We conjecture that the canonical components for the links obtained by 1/n Dehn filling on one component of the 3-chain link are all rational surfaces isomorphic to P^2 blown up at 9n + 1 points. A major goal is to understand how the algebro-geometric structure of these varieties reflects the topological structure of the associated manifolds. At the end of Chapter 3 we discuss common features of these examples and explain how our results lend insight into the affect Dehn surgery has on the character variety. We conclude, in Chapter 4, with a description of possible directions for future research.

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