Universal 2-bridge knot and link orbifolds

Abstract

Let (L,n) be the orbifold with singular set a nontoroidal 2-bridge knot or link L in S3, with cyclic isotropy group of order n. The authors show that the orbifold fundamental group Γ=π1(L,12n) is universal: Γ is isomorphic to a discrete group of isometries of the hyperbolic 3-space H3, and any closed oriented 3-manifold is homeomorphic to H3/G for some subgroup of finite index G of Γ. They show that the Borromean link in S3 is a sublink of the preimage of the singular set of a branched cover over L, with branching indices dividing 12. Since they had proved in an earlier paper that the orbifold with singular set the Borromean link and cyclic isotropy groups of orders 4,4,4 is universal, the result follows. In particular, if L is the figure eight knot, then π1(L,12) is both universal and arithmetic.