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Thesis/Dissertation
Date
2023
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Mathematics
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http://dx.doi.org/10.34944/dspace/8563
Abstract
An overarching goal for studying hyperbolic links is to relate the geometric properties of a link’s complement to the combinatorics of its diagrams. Links which admit an alternating diagram, alternating links, are especially useful for this study. Alternating links have traditionally been considered with projection diagrams on the 2-sphere and with complements in the 3-sphere but there is a strong (and growing) field of work on generalizations of hyperbolic alternating links to broader classes of projection surfaces and complements in more general 3-manifolds. This thesis research lies within that setting, with a focus on the right-angled structure, totally geodesic surfaces, commensurability, and arithmeticity of a family of links in thickened surfaces.
We consider the geometry of a class of hyperbolic link complements in thickened surfaces built from Euclidean and hyperbolic tilings. Such a link has right-angled structure if its complement admits a decomposition into generalized polyhedra with all right-angles that glue to form the complete hyperbolic structure on the complement. Champanerkar, Kofman, and Purcell conjecture that there are no right-angled knots in S^3. We show that this conjecture does not extend to the setting of thickened surfaces by constructing an example.
A surface S in hyperbolic 3-manifold M is totally geodesic if any geodesic tangent to S in M is contained in S. Classifying the presence of totally geodesic surfaces in hyperbolic 3-manifolds is a current open problem. Generalizing Gan’s work in the 3-sphere, we define a link to be right-angled generalized completely realizable (RGCR) if it has a complement which admits a decomposition into hyperbolic generalized polyhedra with the combinatorics of its checkerboard polyhedra and has right-angled structure. We employ the combinatorics of gluings of generalized hyperbolic bipyramids to prove an equivalence for generalized alternating links in thickened surfaces being RGCR, their complements containing two totally geodesic checkerboard surfaces, their checkerboard surfaces each containing one type of polygon, and the links having diagrams which satisfy a set of restrictions. We then use these diagram restrictions to find bounds on the number of RGCR links or tilings corresponding to RGCR links, in terms of g for each genus g projection surface. Two manifolds, M and M', are commensurable if they share a finite-sheeted cover. We show that RGCR links corresponding to equivalent tilings are commensurable and consider the arithmeticity of RGCR links.
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