HIDDEN SYMMETRIES AND THE STRUCTURE OF HYPERBOLIC KNOT AND LINK COMPLEMENTSMondal, Priyadip (2021) HIDDEN SYMMETRIES AND THE STRUCTURE OF HYPERBOLIC KNOT AND LINK COMPLEMENTS. Doctoral Dissertation, University of Pittsburgh. (Unpublished)
AbstractAn isometry h between two finite degree covers of a hyperbolic 3-manifold M is called a hidden symmetry of M if h is not a lift of any self-isometry of M. In 1992, Neumann and Reid asked whether there exists a hyperbolic knot other than the figure eight knot and the two dodecahedral knots of Aitchison and Rubinstein whose complement has hidden symmetries. This thesis aims to study hidden symmetries through the lens of this question. We study geometrically converging families of hyperbolic knots obtained by Dehn filling all but one cusp of a link complement and investigate the existence of hidden symmetries of the complements of such knots. We first concentrate on Dehn fillings of three 2-component hyperbolic links and use geometric isolation property for cusps of the links for our study. This portion is mostly based on joint work with Eric Chesebro and Jason DeBlois. We next discuss an effectivization result related to hidden symmetries for one such link and show some relations between the various number fields for these orbifolds. This portion is based on joint work with Eric Chesebro, Jason DeBlois, Neil R Hoffman, Christian Millichap and William Worden. Finally, the thesis investigates hidden symmetries through analysis of certain horoball packings of hyperbolic three space H^3 and related circle packings of C. We show that the existence of hidden symmetries in infinitely geometrically converging knots obtained by Dehn filling all but one cusp of a hyperbolic link will necessitate the existence of certain order 3 symmetries of such circle packings. We implement this result into SnapPy/Python code which can rule out cases not having these kind of symmetries. We use this code to study some links (and in the process some potential non-links) in the tetrahedral census of Fominykh, Garoufalidis, Goerner, Tarkaev and Vesnin and show that except for a few cases, these symmetries do not exist, hence proving that for most of the cases we test, geometrically converging families of knots obtained from Dehn fillings all but one cusp can only contain finitely many elements with hidden symmetries. Share
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