We establish a pair of criteria for proving that most knots obtained by Dehn surgery on a given two-component hyperbolic link lack hidden symmetries. To do this, we use certain rational functions on varieties associated to the link. We apply our criteria to show that among certain infinite families of knot complements, all but finitely many members lack hidden symmetries.A longstanding question in the study of hyperbolic 3-manifolds asks which hyperbolic knot complements, the 3-manifolds obtained by removing a knot from S 3 , have hidden symmetries [37, p. 307]. More recent work of Reid-Walsh [43] and Boileau-Boyer-Cebanu-Walsh [3] relates this to [43, Conjecture 5.2], on commensurability classes of knot complements. We find the original question intriguing simply because hyperbolic 3-manifolds with hidden symmetries are quite common -each manifold that non-normally covers another has them -but hyperbolic knot complements with hidden symmetries seem quite rare. In fact only three are known to have hidden symmetries, a great many are known not to, and no new examples have been found since the publication of [37]. Indeed, the authors Neumann and Reid of [37] later conjectured that no hyperbolic knot complement in S 3 has hidden symmetries, beyond the three already known [16, Problem 3.64(A)].The totality of evidence for this conjecture would still seem to allow for reasonable doubt. Hidden symmetries can be ruled out for (almost) any particular knot complement by straightforward computations using SnapPy [12] and Sage [14] (or Snap, see [11]). For instance, amongst the 300, 000-odd knot complements with at most 15 crossings, only that of the figure-8 has hidden symmetries. Existing tools are harder to apply to families of knot complements and we only know the following classes to lack hidden symmetries: the two-bridge knots other than the figure -8 [43]; the (−2, 3, n)-pretzels [29]; knots obtained from surgery on the Berge manifold [22], [26]; and certain highly twisted pretzel knots with at least five twist regions [34, Prop. 7.5] 1 . Works of Hoffman [25], Boileau-Boyer-Cebanu-Walsh [4], and Millichap-Worden [35] also bear on the question from other directions.Here we provide a plethora of new classes by giving a method for quickly showing that the generic member of certain families of knot complements produced by hyperbolic Dehn surgery lacks hidden symmetries. We were partly inspired to this by [34, Prop. 7.5] and the proof of [22, Theorem 1.1], which are more specific results in the same direction, but we develop a new tool based on the following motto:The cusp parameters of knot complements obtained from a given hyperbolic link complement M by hyperbolic Dehn filling are recorded by rational functions on the character or deformation variety of M that are smooth near the complete structure.
Neumann and Reid conjecture that there are exactly three knot complements which admit hidden symmetries. This paper establishes several results that provide evidence for the conjecture. Our main technical tools provide obstructions to having infinitely many fillings of a cusped manifold produce knot complements admitting hidden symmetries. Applying these tools, we show for any two-bridge link complement, at most finitely many fillings of one cusp can be covered by knot complements admitting hidden symmetries. We also show that the figure-eight knot complement is the unique knot complement with volume less than 6v0 ≈ 6.0896496 that admits hidden symmetries. We then conclude with two independent proofs that among hyperbolic knot complements only the figureeight knot complement can admit hidden symmetries and cover a filling of the two-bridge link complement S 3 \ 6 2 2 . Each of these proofs shows that the technical tools established earlier can be made effective.
Neumann and Reid conjectured that only three hyperbolic knot complements admit hidden symmetries. Here, we provide evidence for the conjecture, giving obstructions for a manifold to have infinitely many fillings that are knot complements with hidden symmetries. Applying these, we show that at most finitely many fillings of any hyperbolic two-bridge link complement can be covered by knot complements with hidden symmetries. We then make our tools effective, showing first that the only knot complement with hidden symmetries and volume less than $6v_0 \approx 6.0896496$ is the complement of the figure-eight. We conclude with two proofs that if a hyperbolic knot’s complement admits hidden symmetries and covers a filling of the complement of the $6_2^2$ link, it is the figure-eight.
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