Universal 2-bridge knot and link orbifolds



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Hilden, Hugh Michael and Lozano Imízcoz, María Teresa and Montesinos Amilibia, José María (1993) Universal 2-bridge knot and link orbifolds. Journal of Knot Theory and Its Ramifications, 2 (2). pp. 141-148. ISSN 0218-2165

Official URL: http://www.worldscientific.com/doi/abs/10.1142/S021821659300009X


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.

Item Type:Article
Uncontrolled Keywords:arithmetic fundamental group; orbifold; cyclic isotropy; singular set; nontoroidal 2-bridge knot; fundamental group; discrete subgroup of hyperbolic isometries of hyperbolic 3-space
Subjects:Sciences > Mathematics > Topology
ID Code:22171
Deposited On:01 Jul 2013 17:11
Last Modified:12 Dec 2018 15:13

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