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Geometry and Topology | Mathematics


In this paper, we explicitly construct large classes of incommensurable hyperbolic knot complements with the same volume and the same initial (complex) length spectrum. Furthermore, we show that these knot complements are the only knot complements in their respective commensurability classes by analyzing their cusp shapes.

The knot complements in each class differ by a topological cut-and-paste operation known as mutation. Ruberman has shown that mutations of hyperelliptic surfaces inside hyperbolic 3-manifolds preserve volume. Here, we provide geometric and topological conditions under which such mutations also preserve the initial (complex) length spectrum. This work requires us to analyze when least area surfaces could intersect short geodesics in a hyperbolic 3-manifold.

Document Type

Published Version


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First published in Communications in Analysis and Geometry in volume 25, issue 3, published by International Press.

Original Citation

Christian Millichap
Mutations and short geodesics in hyperbolic 3-manifolds.
Communications in Analysis and Geometry, 2017, volume 25, issue 3, pages 625-683



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