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Hidden symmetries of hyperbolic links
Han Yoshida

National Institute of Technology, Nara College

1Introduction

Hidden symmetry of a manifold M is a homeomorphism of finite degree covers of M that

does not descend to an automorphism of M . In [7], W. Neumann and A. Reid conjectured
that the figure‐eight knot and the two dodecahedral knots are the only hyperbolic knotsin S^{3} admitting hidden symmetries. Many researchers concerned with this problem. M.

Macasieb, and T. W. Mattman [5] showed that (-2,3, n) pretzel knot (n\in \mathbb{N}) does not
admit hidden symmety. By using computer, O. Goodman, D. Heard and C. Hodgson [3]have verified for hyperbolic knots with 12 or fewer crossings. A. Reid and G. S. Walsh [8]showed that non‐arithmetic 2‐bridge knot complements admit no hidden symmetry.

For two component links, E. Chesebro and J. DeBlois [1] constructed infinitely many
two components non‐arithmetic link complements admitting hidden symmetries. Let
C_{i} (i=1 , 3) be links as in Figure 1. O. Goodman, D. Heard and C. Hodgson showedthat S^{3}-C_{2} and S^{3}-C_{3} have non‐trivial hidden symmetries by using computer. S^{3}-C_{2}
is obtained by cutting along the colored two punctured disk of S^{3}-C_{1} and regluing it.
Repeat this process about the colored two punctured disk of S^{3}-C_{2} . We can obtainS^{3}-C_{3} . J. S. Meyer, C. Millichap and R. Trapp [6] constructed n(\geq 6) component link

C1 C2 c_{3}Figure 1: 5‐link chain.

complements admitting hidden symmetries. They prove this by analyzing of the geometryof these link complements, including their cusp shapes and totally geodesic surfaces insideof these manifolds.

In this paper, we generalize the result of O. Goodman, D. Heard and C. Hodgson. LetL be ann+1 ‐component alternating chain link as in the left side of Figure 2(n\geq 4) .

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Cut along the colored two punctured disk of S^{3}-L and reglue it. We name the resultingn ‐component link L_{n}.
Theorem 1 S^{3}-L_{n} is non‐arithmetic and admits a hidden symmetry (n\geq 4) .

c_{1}LL_{n}Figure 2: The link L_{n}.

We prove this theorem by using ideal polyhedral tessellation of \mathbb{H}^{3}.
2

Commensurator and normalizer

Two subgroups G_{1}, G_{2}<Isom(\mathbb{H}^{3}) are said to be commensurable if and only if theirintersection G_{1}\cap G_{2} has finite index in both G_{1} and G_{2}.

G_{1} and G_{2} are said to becommensurable in the wide sense if and only if there is a h\in Isom(\mathbb{H}^{3}) such that G_{1}is commensurable with h^{-1}G_{2}h . The notion of commensurability can be directly trans‐ported to hyperbolic orbifolds by considering the respective fundamental groups. Then,commensurable hyperbolic orbifolds admit a finite‐sheeted common covering orbifold.
Commensurability is an equivalence relation.
For a Kleinian group \Gamma , the commensurator of \Gamma is defined byComm (\Gamma)= { g\in Isom(\mathbb{H}^{3}) : g\Gamma g^{-1} and \Gamma are commensurable.}.
Clearly, Comm (\Gamma)>\Gamma . Let \Gamma be a finitely generated Kleinian group of finite co‐volume. It

is well known that Comm ( \Gamma ) is a commensurabilty invariant (see [10]). Comm ( \Gamma ) contains
every member of the commensurability class. G. Margulis [4] showed that Comm ( \Gamma ) is
discrete if and only if \Gamma is non‐arithmetic. For a non‐arithmetic group \Gamma, Comm(\Gamma)
contains every member of the commensurability class “in finite index”
The normalizer of \Gamma is

N(\Gamma)=\{g\in Isom(\mathbb{H}^{3})):g\Gamma g^{-1}=\Gamma\}.
Clearly, N(\Gamma)<Comm(\Gamma) . N(\Gamma)/\Gamma\simeq Isom(\mathbb{H}^{3}/\Gamma) and N(\Gamma) is discrete.

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31If N(\Gamma)\neq Comm(\Gamma) , we say \Gamma admits a hidden symmetry. For an arithmetic Kleiniangroup \Gamma, Comm(\Gamma) is not discrete. Thus arithmetic Kleinnian group always admits ahidden symmetry.
3

Proof of Main Theorem

Let \Gamma (resp. \Gamma_{n} ) be a Kleinian group such that S^{3}-L=\mathbb{H}^{3}/\Gamma (resp. S^{3}-L_{n}=\mathbb{H}^{3}/\Gamma_{n} ).
W. Neumann and A. Reid showed that S^{3}-L is non‐arithmetic (see [7] Theorem 5.1).
W. Thurston [9] showed S^{3}-L is obtained by glueing two ideal drums as in Figure 3.
The side angles of this drum are\arccos(\frac{\cos\pi/(n+1)}{\sqrt{2} ) and other angles are\pi-2\alpha.

The colored two punctured disc corresponds to the colored ideal quadrilaterals as inFigure 3. S^{3}-L_{n} is obtained by cutting along the colored two punctured disk and reglueing
it. Thus, S^{3}-L_{n} is obtained by glueing two ideal drums as the arrows are matched asin Figure 3. Lift the ideal polyhedral decompositions of S^{3}-L and S^{3}-L_{n} . We can5^{-}\ltimes\sim

2^{-}\ltimes\sim

u_{A \frac{1}2 \sim}^{1 4|_{\backsl h_{1-2}^{\backsl h}|_{\backsl h _{J}\varsigmaj.A\backsl h_{-2}\per 35I- \backslash , \backu\srflolar^{As_-h}^{\ba1_ckslah{_{-152-2}}|^\{\bapeckslahr} pjA\^ove{rli\ne{\omelfo r_{\backgslaah}}3\perp 4I

J_{\backslash_{/}^{\triangle ft}\frac{1}\xi}b\sim\backslash_{-2} \perp^{4I}513\frac{1^2} \backslh_{5^\backslh_{1-2}^A\overlin{backslh\prime}-tanglriht_{\omegasi}\bcklash!^{\timesbgwde_{-2}\per^{4lconer}|_{s13u\lfo r_{x_{1- 2}5_{\overline{N_{S}^{\ltimes\bigwedge_{\wedge}S^{3_{-\llcorner}} S^{3}-L_{n}Figure 3: Ideal polyheral decompositions of S^{3}-L and S^{3}-L_{n}(n=4) .

get the same ideal polyhedral tessellation of \mathbb{H}^{3} . Denote it by T . The symmetry group ofT is discrete. \Gamma and \Gamma_{n} preserve the tessellation T . Thus \Gamma and \Gamma_{n} are commensurablewith the symmetry group of T . As commensurability is an equivalence relation, \Gamma_{n} iscommensurable with the non‐arithmetic group \Gamma . Hence, Comm (\Gamma_{n})= Comm ( \Gamma) and \Gamma_{n}is non‐arithmetic.

Let P be an ideal drum which is a lift of this ideal polyhedral decomposition. We
consider the symmetry that rotates the chain L clockwise, taking each link into the next.
This corresponds to 2\pi/(n+1) ‐rotation about the geodesic which is perpendicular tothe top and bottoms of P . Denote it by \gamma . As \gamma is a lift of a symmetry of \mathbb{H}^{3}/\Gamma,\gamma\in N(\Gamma)< Comm (\Gamma)=Comm(\Gamma_{n}) .
Let c_{1} ,, c_{n+1} (resp. cí. , c_{n}' ) be the cusps of S^{3}-L (resp. S^{3}-L_{n} ) as in Figure

2. The cusp c_{i} corresponds to two ideal vertices of P . By cuttng and re‐glueing along thecolored twice punctured disk, c_{2} and c_{n+1} correspond to the cusp c_{2}' . (See Figure 3.)

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Let V_{i} be the ideal vertices of \mathbb{H}^{3} which correspond to the cusp c_{i} We can see \gamma(V_{1})\neq V_{i}(i=1 , n-1) . \gamma is not a lift of isometry of \mathbb{H}^{3}/\Gamma_{n} . Thus \gamma\not\in N(\Gamma_{n}) .
We have N(\Gamma_{n})\neq Comm(\Gamma_{n}) . Hence \Gamma_{n} admits a hidden symmetry.

Figure 4: Rotation of S^{3}-L(n=4) .

References[1] E. Chesebro, J. DeBlois Hidden symmetries via hidden extensions,
https://arxiv.org/pdf/1501.00726.pdf[2] D. Epstein, R. Penner, Euclidean decompositions of noncompact hyperbolic mani‐folds, J. Differential Geom. 27 (1988), 67‐80.

[3] O. Goodman, D. Heard, C. Hodgson. Commensurators of cusped hyperbolic mani‐folds, Experiment. Math. Volume 17, Issue 3 (2008) 283‐306.
[4] G. Margulis. Discrete Subgroups of Semi‐simple Lie Groups, Ergeb. der Math. 17,Springer‐Verlag (1989).
[5] M. Macasieb, T. W. Mattman, Commensurability classes of (−2;3;n) pretzel knot
complements. Algebr. Geom. Topol., 8(3):(2008) 1833−1853[6] J. S. Meyer, C. Millichap, R. Trapp Arithmeticity and hidden symmetries of fully
augmented pretzel link complements, https://arxiv.org/pdf/1811.00679[7] W. D. Neumann and A.W. Reid. Arithmetic of hyperbolic manifolds. In TOPOLOGY
90, Proceedings of the Research Semester in Low Dimensional Topology at Ohio

State University. De Gruyter Verlag (1992) 273‐310.
[8] A. W. Reid, G. S. Walsh. Commensurability classes of 2‐bridge knot complements.
Algebr. Geom. Topol., 8(2):(2008) 1031−1057[9] W. P. Thurston, The geometry and topology of three‐manifolds, Lecture notes,
Princeton University, 1976‐80.

33[10] G. S. Walsh, Orbifolds and commensurability, https://arxiv.org/pdf/1003.1335.pdf.
National Institute of Technology, Nara College
22 Yatamachi, Yamatokoriyama Nara 639‐1080
JAPAN

E‐mail address: han@libe.nara‐k.ac.jp