Arithmetic Knots in Closed 3-Manifolds

Abstract

Let Od denote the ring of integers in [Formula: see text]. An orientable finite volume cusped hyperbolic 3-manifold M is called arithmetic if the faithful discrete representation of π1(M) into PSL(2,C) is conjugate to a group commensurable with some Bianchi group PSL (2,Od). If M is a closed orientable 3-manifold, we say a link L ⊂ M is arithmetic if M\L is arithmetic. In the paper, we show that there exist closed orientable 3-manifolds which do not contain anarithmetic knot. Our methods give much more precise informations for non-hyperbolic 3-manifolds. For certain Lens Spaces, we can give fairly complete statements.

The volume conjecture, perturbative knot invariants, and recursion relations for topological strings

“…This correspondence is checked explicitly for two examples, the figure eight knot complement and the once punctured torus bundle over S 2 with the holonomy L 2 R, which is isomorphic to the SnapPea census manifold m009 [10], which is the complement of a knot in a three-manifold of different topology than the three-sphere [11,12,13]. Up to subleading order, one can find coincidence for these examples.…”

Jørgensen number and arithmeticity

“…K minimum polynomial of z numerical value of z 5 2 1 + 2z + z 2 + z 3 −0.21507985 + 1.307141279i 6 1 1 − 2z + 3z 2 − z 3 + z 4 0.104876618 − 1.552491820i 7 4 1 + 4z − 4z 2 + z 3 2.10278472 + 0.665456952i 7 7 1 − z + 3z 2 − 2z 3 + z 4 0.95668457 − 1.227185638i…”

Counting cusps of subgroups of $\mathrm {PSL}_2(\mathcal {O}_K)$

“…The fundamental group of the complement of the figure-eight knot in S 3 injects as an index 12 subgroup containing Γ(4) (see [4]). The fundamental group of the sister of the figure-eight knot complement, a knot in the lens space L(5, 1), injects as an index 12 subgroup containing Γ(2) (see [1]). Reid [10] has shown that the figure-eight knot complement is the only arithmetic knot complement in (The fundamental groups of cyclic covers of the figure-eight knot complement all have one cusp.)…”