Two-bridge links, pretzel knots and bi-orderability




Johnson, Jonathan C., Ph. D.

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The orderability of a 3-manifold group is closely connected to the topological properties of the manifold. Link groups are always left-orderable. However, there are link groups which are known to be bi-orderable, as well as link groups known not to be bi-orderable. In this dissertation, the bi-orderability of some families of link groups is shown. We show that two-bridge links with Alexander polynomials whose coefficients are coprime are extensions of ℤ by residually torsion-free nilpotent groups. It follows from a result of Linnell-Rhemtulla-Rolfsen[30] that if a two-bridge link has an Alexander polynomial with coprime coefficients and all real positive roots, then its link group is bi-orderable. In particular, if a two-bridge knot has an Alexander polynomial with all real positive roots, then its knot group is bi-orderable. This result shows that a large family of knots whose cyclic branched covers are known to be L-spaces have bi-orderable knot groups. Additionally, using a technique developed by Mayland, the pretzel knots P(-3, 3, 2r + 1) are shown to have bi-orderable knot groups. Issa-Turner[20] showed that all the cyclic branched covers of these knots are L-spaces. Finally, a family of genus one pretzel knots are shown to have bi-orderable knot groups and double branched covers which are not L-spaces.



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