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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 23rd 2019
    • (edited Dec 23rd 2019)

    Naive question:

    The complement of the unknot in the 3-sphere admits a hyperbolic metric, namely the Euclidean BTZ black hole (explicit e.g. in Gukov 03, Appendix A).

    But Thurston 82, Cor. 2.5 states that the complement of the unknot does not have “hyperbolic structure”.


    How is that not a contradiction? Is there some asymptotic condition which is violated?

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 24th 2019

    Thurston is surely right about the solid torus not being hyperbolic. So what is Gukov saying? It’s the quotient of hyperbolic 3\mathbb{H}^3, but is it by a discrete, torsion-free group of isometries?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 24th 2019

    Thanks. Right, it must be some technical fine-print like this. I’ll try to check, but have to run now.

    But, by the way, “hyperbolic solid torus” is a common term used by many authors, also in maths…

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 24th 2019

    Domenico found it: The condition violated by the hyperbolic solid torus is: finite volume. (

    Now that makes me wonder: Do all torus knots have hyperbolic complements if we allow infinite volume?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 25th 2019

    The result of the original articles on Euclidean BTZ black holes

    is that, generally they are quotients of hyperbolic 3-space by Schottky groups, and more generally, if they are rotating, by quasi-Fuchsian groups which are Fenchel-Nielsen deformations of these Schottky groups.

    Question: Which of these quotients are knot complements?