key: cord-0583028-m2gz0ptx
authors: Ueki, Jun
title: Modular knots obey the Chebotarev law
date: 2021-05-22
journal: nan
DOI: nan
sha: 80e16ccd1bf4cf142aa7316ae1a7c8fd8d8ba2aa
doc_id: 583028
cord_uid: m2gz0ptx

We refine McMullen's construction of sequences of knots in $S^3$ obeying the Chebotarev law in two ways. One is to extend his theorem for generalized pseudo-Anosov flows, which may admit a finite number of 1-pronged singular orbits. The other is to invoke the notion of rational Fried surgeries to produce many pseudo-Anosov flows. These methods especially prove that modular knots around the missing trefoil in $S^3$ obey the Chebotarev law, yielding an equidistribution theorem for the Rademacher symbol.

A Chebotarev link in S 3 is an analogue of the set of all prime numbers in Z, and would play an important roll in arithmetic topology, especially when we formulate an analogue of the idelic class field theory for 3-manifolds [Uek21] (see also [Mor12, Nii14, NU19, Mih19, KMNT21] ). Here is the definition:

Definition 1 (The Chebotarev law). Let (K i ) = (K i ) i∈N >0 be a sequence of disjoint knots in a 3-manifold M . For each n ∈ N >0 and j > n, put L n = ∪ i n K i and let [K j ] denote the conjugacy class of K j in π 1 (M − L n ). We say that (K i ) obeys the Chebotarev law if the density equality lim ν→∞ #{n < j ν | ρ([K j ]) = C} ν = #C #G holds for any n ∈ N >0 , any surjective homomorphism ρ : π 1 (M − L n ) → G to any finite group, and any conjugacy class C ⊂ G. A countably infinite link K is said to be Chebotarev if it obeys the Chebotarev law with respect to some order.

In order to answer Mazur's question on the existence of a Chebotarev arrangement in [Maz12] , McMullen proved a highly interesting theorem, generalizing Adachi-Sunada's result [Sun84, , by using Parry-Pollicott's theory on zeta functions of symbolic dynamics [PP90]:

Proposition 2 [McM13, Theorem 1.2]. Let (K i ) be the closed orbits of any topologically mixing pseudo-Anosov flow on a closed 3-manifold M , ordered by length in a generic metric. Then (K i ) i obeys the Chebotarev law. (We consult [Mos92, Fen08, Cal07] for the terminology of pseudo-Anosov flows.) By applying his theorem to the monodromy suspension flow of the figure-eight knot K and noting that the Chebotarev law persists under Dehn surgeries, he constructed a Chebotarev link containing K in S 3 [McM13, Corollary 1.3], claiming that his construction is applicable for any hyperbolic fibered knot. In this article, we refine his construction in two ways to verify his claim for some subtle cases (see Section 2), as well as to improve the author's argument in the previous paper [Uek21, Propositions 12, 15] . Namely, we first aim to verify the following assertions.

Theorem 3. Let L be a fibered hyperbolic link in S 3 and let (K i ) denote the sequence of knots consisting of the closed orbits of the suspension flow of the monodromy map and L itself. Then (K i ) obeys the Chebotarev law, if ordered by length with respect to a generic metric.

The union L = ∪ i K i is a stably Chebotarev link, that is, for any finite branched cover h : M → S 3 branched along any finite link in L, the inverse image h −1 (L) is again Chebotarev.

One way is to extend McMullen's theorem for generalized pseudo-Anosov flows, which allow 1-pronged singular orbits (Section 3). The other is to invoke the notion of rational Fried surgeries, which produce many (generalized) pseudo-Anosov flows (Section 4).

We remark that the Chebotarev link L in Theorem 3 is also called the planetary link rising from L and sometimes admits another amazing property due to Ghrist and others; the link L may contain every type of links. A fused question is attached in Section 5.

Our refinement further provides a new example, which also answers Mazur's question:

Theorem 4. Modular knots and the missing trefoil in S 3 obey the Chebotarev law, if ordered by length in a generic metric.

As a corollary, we obtain an equidistribution formula for a certain ubiquitous function called the Rademacher symbol Ψ : SL 2 Z → Z (Corollary 9). Details are given in Sections 6 and 7.

Let us recall McMullen's construction; The mapping torus of A = 1 1 1 2 with tr A > 2 acting on T = R 2 /Z 2 is the result K(0) of the 0-surgery along the figure-eight knot K = 4 1 in S 3 [BZH14, Chapter 5] , so that the monodromy suspension flow is a pseudo-Anosov flow on the torus bundle K(0) such that the exterior of the 0-orbit is homeomorphic to S 3 − K. By [McM13, Corollary 2.2], the flow is topologically mixing if there are two orbits K i and K j whose length satisfy ℓ(K i )/ℓ(K j ) ∈ Q, hence a generic time change makes the pseudo-Anosov flow to be topologically mixing. Therefore by [McM13, Theorem 1.2], the closed orbits in K(0) obey the Chebotarev law. Since the Chebotarev law persists under Dehn surgeries along knots, the ∞-surgery along the 0-orbit yields a sequence of knots in S 3 containing K itself and obeying the Chebotarev law.

Here we attach a proof of a basic fact used in the construction:

Proposition 5. The Chebotarev law persists under Dehn surgeries along knots.

Proof. Let (K i ) i be a sequence of disjoint knots in M obeying the Chebotarev law and let M ′ denote the result of Dehn surgeries along a link contained in L N for N ∈ Z >0 . If N n, then we have M − L n = M ′ − L n , hence the density equality persists. If N > n, consider the composition

for any j N , the density equality for ρ yields that for ρ.

If we try exactly the same construction for any other hyperbolic fibered knot or link in S 3 , we should notice that the 0-surgery is not always hyperbolic. If the result of 0-surgery is hyperbolic, then in general we do not necessarily obtain a pseudo-Anosov flow on a closed 3-manifold.

Indeed, Gabai [Gab97, p.27] initially pointed out by using Penner's program that the 0-surgery along K = 8 20 results a fibered manifold with reducible monodromy, so that the mapping torus is toroidal, hence not hyperbolic. This fact is known as the consequence of the fact that the complement of 8 20 had an essential punctured torus with boundary slope 0 [HO89, p.479]. Here for the convenience of readers, we give a slightly generalized assertion (Proposition 6) suggested by Motegi. Note that K = 8 20 is the pretzel knot P (2, 3, −3) and its crosscap number is two.

Proposition 6. The 0-surgery along a hyperbolic knot K with the crosscap number two in S 3 results in a toroidal 3-manifold.

Proof. Let IntV K denote the interior of a tubular neighborhood of K. Then in the exterior S 3 − IntV K of K, we have a once punctured Klein bottle Σ whose boundary is a preferred longitude of K. A tubular neighborhood of Σ in the exterior is a twisted I-bundle over Σ, and the ∂I-subbundle is a twice punctured torus with its boundary components again being preferred longitudes of K. Hence in the result K(0) of the 0-surgery along K, we have a twisted I-bundle X over a Klein bottle Σ, in which the ∂I-subbundle is an incompressible torus T .

Consider the decomposition

In the former case, π 1 (K(0)) is a finite group, contradicting H 1 (K(0)) ∼ = Z. In the latter case, K(0) is reducible, contradicting Gabai's great result [Gab87, Corollary 8.3 ]. If Y is reducible, then Y is the connected sum V #Z of a solid torus V and some closed 3-manifold Z = S 3 . Hence we have K(0) = W #Y for a prism manifold W or K(0) = RP 3 #RP 3 #Z, again contradicting Gabai's result.

Examples of hyperbolic fibered knots such that the 0-surgeries yield Seifert fibered spaces with periodic monodromies were given in [MS05] (see also [IM09] ).

In addition, hyperbolic surgeries do not always result Seifert fibered spaces, so that we do not necessarily obtain a pseudo-Anosov flow on a S 1 -bundle over a closed surface.

Recall that a pseudo-Anosov flow admits the stable/unstable foliations with finite number of kpronged singular orbits with k 3 and away from them it is just an Anosov flow. A generalized pseudo-Anosov flow, which is also called a 1-pronged pseudo-Anosov flow or a singular Anosovflow, may admit finite number of 1-pronged singular orbits and away from them it is a pseudo-Anosov flow. Since the whole argument for pseudo-Anosov flows in [Col79] applies, a generalized pseudo-Anosov flow still admits a Markov section, so that the proof of [McM13, Theorem 1.2] (contained in Proof of Theorem 1.4) works as well, yielding the following assertion.

Theorem 7. Let (K i ) i be the closed orbits of any topologically mixing generalized pseudo-Anosov flow on a closed 3-manifold M , ordered by length in a generic metric. Then (K i ) i obeys the Chebotarev law.

This theorem together with Proposition 5 proves Theorem 3;

Proof of Theorem 3. Let L in S 3 be any fibered hyperbolic link other than 4 1 . Then Nielsen-Thurston's classification together with Thurston's theorem [Thu86, Proposition 2.6, Theorem 0.1] yield that the exterior S 3 − L is the Mapping torus of a pseudo-Anosov map on a punctured surface and that the monodromy suspension flow is pseudo-Anosov.

A pseudo-Anosov map on a punctured surface Σ extends to a map on a closed surface Σ such that the stable/unstable foliations may admit 1-pronged singular closed orbits [Thu88, CB88, FM12] . The pseudo-Anosov suspension flow on the mapping torus of Σ extends to a generalized pseudo-Anosov flow on that of Σ such that the components of L are contained in the set of the closed orbits. By [McM13, Corollary 2.2] based on [PP90, Theorem 8.5], a generic time change makes the flow topologically mixing. Hence Theorem 7 yields a sequence of knots obeying the Chebotarev law.

Applying Proposition 5 for the ∞-Dehn surgeries along L in the mapping torus of the closed surface Σ, we obtain a sequence of knots in S 3 obeying the Chebotarev law, proving the first assertion of Theorem 3.

Since a generalized pseudo-Anosov flow lifts to that on a finite branched cover branched along a finite set of closed orbits, we clearly obtain the second assertion of Theorem 3.

Fried surgeries provide alternative proof of Theorem 3.

Let K be a closed orbit of an Anosov flow and σ a cross section σ in the boundary of the tubular neighborhood of K such that σ transversely intersects each of the stable/unstable flow lines twice. Then a Fried surgery along (K, σ) naturally induces an Anosov flow on the result of the Dehn surgery whose slope coincides with that of σ. It is defined as the combination of the blow-up along K and the blow-down by parallel copies of σ, so that the orbits of the flow and the stable/unstable foliations persist in the exterior [Fri83] . We remark that Fried surgeries are topologically equivalent to Goodman surgeries [Goo83, Sha20].

If we instead apply a similar procedure for a closed orbit of a generalized pseudo-Anosov flow and a cross section σ such that σ transversely intersects each of the stable/unstable flow lines more than once, then we mostly obtain a k-pronged orbit with k 2 and rarely a 1-pronged orbit in a new flow on the result of the corresponding rational Dehn surgery. Note in addition that all but finite rational surgeries are hyperbolic by Thurston [Thu82] . Therefore, Theorem 3 is verified also by combining the 0-fillings, rational Fried surgeries, a generic time change, [McM13, Theorem 1.2], the ∞-Dehn surgeries, and Proposition 5. This procedure avoids the use of Theorem 7.

If the meridians of the link L are not parallel to the stable/unstable flow lines, then the ∞-Fried surgery is defined, so that we may directly apply [McM13, Theorem 1.2] or Theorem 7 to the result of the ∞-Fried surgery on the result of the 0-filling, to obtain Theorem 3. For instance, the figure-eight knot is not the case. Indeed, the ∞-blow down yields four singular points on the figure-eight knot, two of them are attracting and the other two are repelling.

Even if the ∞-Fried surgeries are not defined, we may still play an analogue of number theory on the foliated dynamical systems by adding Reeb components, as described in [KMNT21].

We remark that the Chebotarev link L = ∪ i K i obtained by Theorem 3 is called the planetary link rising from L, after Birman-Williams [BW83] . Ghrist and others proved that if L belongs to a certain large class of links (eg, the figure-eight knot, the Whitehead link, the Borromean ring, and every fibered non-torus 2-bridge knot), then the planetary link L contains all types of links [Ghr97, GHS97, GK04] .

McMullen's construction [McM13] answers Mazur's question [Maz12] on the existence of a sequence of knots in S 3 obeying the Chebotarev law. Morishita further asks the following:

Question 8. Is there exist a sequence of disjoint knots (K i ) i in S 3 obeying the Chebotarev law such that every type of knot appears in the sequence exactly once?

In order to answer his fused question in affirmative, it suffices to show the equidistribution of each type of knot in the planetary link. Miller's work [Mil01] might give a cliff, which explicitly finds some types of knots in the planetary link. We leave this question for a future study.

Let H 2 = {z ∈ C | Imz > 0} denote the upper half plane. The unit tangent bundle of the modular orbifold PSL 2 Z \ H 2 is well-known to be homeomorphic to both the quotient space PSL 2 Z \ PSL 2 R ∼ = SL 2 Z \ SL 2 R and the exterior of a trefoil K in S 3 . A flow on PSL 2 Z \ PSL 2 R historically called the geodesic flow is defined by multiplying e t 0 0 e −t on the right, and its closed orbits are called modular knots.

An element γ ∈ SL 2 Z is said to be hyperbolic if | tr γ| > 2 holds. For each primitive hyperbolic element γ, we may define the corresponding modular knot C γ by C γ (t) = M γ e t 0 0 e −t (0

with ξ γ > 1. Every modular knot may be presented by some primitive hyperbolic γ, so that there is a natural surjection from the set of hyperbolic primitive elements to that of modular knots.

The geodesic flow on PSL 2 Z \ PSL 2 R is classically known to be Anosov and its structure is finely studied. After a certain compactification discussed in [BP20], the flow extends to a generalized pseudo-Anosov flow on the result of the 0-filling, in which the missing trefoil becomes a 1-pronged singular closed orbit. Therefore, by our Theorem 7 and Proposition 5, or by some rational Fried surgery and [McM13, Theorem 1.2] together with Proposition 5, we obtain the assertion of Theorem 4 stated in Section 1; modular knots and the missing trefoil in S 3 obeys the Chebotarev law, if ordered by length in a generic metric.

Here we give a remark on Lorenz knots; The classical Lorenz flow in R 3 [Lor63] is defined by

and its closed orbits in R 3 ∪ {∞} is called Lorenz knots. This flow is well-known for its robustness and the chaotic behavior of its orbits, so that it may suggest the existence of "fate" and "butterflyeffects" at the same time. By virtue of Ghys [Ghy07] together with BP20] , based on Birman-Williams' template theory, the set of modular knots and the missing trefoil in S 3 are conjecturally topologically equivalent to that of Lorenz knots. Hence we may currently say that Lorenz knots conjecturally obey the Chebotarev law. We in addition remark that gives a parametrized family of Lorenz-like flows whose closed orbits are topologically equivalent to modular knots and the missing trefoil.

The discriminant function ∆(z) = q ∞ n=1

(1 − q n ) 24 with q = e 2π √ −1z , z ∈ H 2 is a well-known modular function of weight 12. The Dedekind symbol Φ and the Rademacher symbol Ψ are the functions SL 2 Z → Z satisfying the equalities

we take a branch of the logarithm so that −π Im log z < π holds. This Ψ factors through the conjugacy classes of PSL 2 Z. (We may find in many literatures various confusions about the convention of the Rademacher symbol. Our convention is based on Matsusaka's quite thorough investigation; See [Mat20] .)

The Rademacher symbol Ψ is known to be a highly ubiquitous function. Indeed, Atiyah proved the equivalence of seven definitions rising from very distinct contexts [Ati87], whereas Ghys gave further characterizations (cf.[BG92]) especially by using modular knots [Ghy07, Sections 3.3-3.5] (see also [DIT17, Appendix] ), proving that for each primitive hyperbolic γ ∈ SL 2 Z, the linking number between the modular knot C γ and the missing trefoil K coincides with the Rademacher symbol, namely, lk(C γ , K) = Ψ(γ) holds. Note that the length of the image of C γ on the modular orbifold is given by ℓ(γ) = 2 log ξ γ , where ξ γ = | tr γ| + (tr γ) 2 − 4 2 denotes the larger eigenvalue of γ, hence the order defined by the length of C γ in some metric coincides with that defined by |trγ|. This order persists up to finite permutation under the filling and surgeries. Now let (K i ) denote the sequence of modular knots and the missing trefoil ordered by length and suppose that L n = ∪ i n K i contains the missing trefoil K. Let 0 = m ∈ Z. Applying the Chebotarev law for the surjective homomorphism ρ : π 1 (S 3 − L n ) ։ Z/mZ; [C γ ] → lk(K, C γ ) = Ψ(γ) mod m, we obtain the following equidistribution formula:

Corollary 9. Suppose that γ runs through primitive hyperbolic elements of SL 2 Z. For any m ∈ Z >0 and k ∈ Z/mZ, we have the density equality lim ν→∞ #{γ | | tr γ| < ν, Ψ(γ) = k in Z/mZ} #{γ | | tr γ| < ν} = 1 m .

The similar arguments may be applicable to the unit tangent bundles of cusped hyperbolic orbifolds and their covering spaces. For instance, triangle modular knots around any torus knot in S 3 will be finely studied in [MU21] , that would be compared with Dehornoy-Pinsky's works on templates and codings [Deh15, DP18].

We finally remark another distribution theorem due to Sarnak-Mozzochi, proved by using the trace formula;

Proposition 10 [Sar08, Sar10, Moz13] . Suppose that γ runs through conjugacy classes of primitive hyperbolic elements in SL 2 Z. Then for any −∞ a b ∞, we have lim ν→∞ #{γ | ℓ(γ) < ν, a Ψ(γ) ℓ(γ) b} #{γ | ℓ(γ) < ν} =

π .

Exchanging of the order of the orbits is an important issue in the context of dynamical systems. The study of Chebyshev bias [RS94] of prime numbers and prime geodesics is of a similar interest. We wonder whether these two equidistribution formulas in above may be interpreted from a unified viewpoint.

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I would like to express my sincere gratitude to Yuta Nozaki, Hirokazu Maruhashi, and Kimihiko Motegi, and Toshiki Matsusaka to whom I greatly owe the introductions to low-dimensional generalities, foliated dynamics, non-hyperbolic 0-surgeries, and the Rademacher symbols respectively. I also thank Masanori Morishita and Hirofumi Niibo for cheerful communications under the COVID19 situation. I am and will be grateful to the anonymous referees and experts of journal(s) for careful reading and sincere comments. The author has been partially supported by JSPS KAKENHI Grant Number JP19K14538.