Algebraic invariants, mutation, and commensurability of link complements

COMMENSURABILITY CLASSES OF (−2, 3, n) PRETZEL KNOT COMPLEMENTS

Let K be a hyperbolic (−2, 3, n) pretzel knot and M = S \ K its complement. For these knots, we verify a conjecture of Reid and Walsh: there are at most three knot complements in the commensurability class of M . Indeed, if n 6= 7, we show that M is the unique knot complement in its class. We include examples to illustrate how our methods apply to a broad class of Montesinos knots.

Principal Congruence Link Complements

Let d be a square-free positive integer, let Od denote the ring of integers in Q( √ −d), and let Qd denote the Bianchi orbifold H/PSL(2,Od). A non-compact finite volume hyperbolic 3-manifold X is called arithmetic if X and Qd are commensurable, that is to say they share a common finite sheeted cover (see [26] Chapter 8 for more on this). If N is a closed orientable 3-manifold and L ⊂ N a link, ...

Regular Tessellation Link Complements

By a regular tessellation, we mean any hyperbolic 3-manifold tessellated by ideal Platonic solids such that the symmetry group acts transitively on oriented flags. A regular tessellation has an invariant we call the cusp modulus. For small cusp modulus, we classify all regular tessellations. For large cusp modulus, we prove that a regular tessellations has to be infinite volume if its fundament...

Algebraic and Differential Invariants

This article highlights a coherent series of algorithmic tools to compute and work with algebraic and differential invariants.

On the Alexander invariants of hypersurface complements