Olympic links in a Chebotarev link

The Chebotarev law for an infinite link is an equidistribution property about how its components are linked in a group theoretic sense. We overview several properties of a Chebotarev link following the author’s article “Chebotarev links are stably generic”. In addition, we exhibit the density of modulo 2 Olympic links in a Chebotarev link. Анотація. Закон Чеботарева для нескінченних завузлень це властивість рівномірності розподілу того, як кола завузлені з точки зору теорії груп. Ми розглядаємо кілька властивостей завузлення Чеботарева, слідуючи статті автора “Завузлення Чеботарева є стабільно типовими”. Крім того, ми показуємо щільність по модулю 2 «олімпійського завузленнь» в завузленнях Чеботарева.


INTRODUCTION
The analogy between knots and prime numbers was initially pointed out by B. Mazur [13] and developed in a systematic manner by M. Kapranov [9], A. Reznikov [23,24], M. Morishita [18,19] and others. The Chebotarev law for an infinite link is an equidistribution property about how its components are linked in a group theoretic sense. An infinite link in S 3 obeying the Chebotarev law might be a good analogue of the set of all prime numbers in several senses. In this article, we overview several properties of Chebotarev links mainly following the author's article [32] and also exhibit the density of modulo 2 Olympic links in a Chebotarev link as a new example of Chebotarev phenomenon.
We first overview the relationship between two analogues in a 3-manifold of the set of rational prime numbers. We recall the definitions and state that if (K i ) iPN ą0 is a sequence of knots obeying the Chebotarev law in the sense of B. Mazur and C. T. McMullen, then K = Y i K i is a stably generic link in the sense of T. Mihara. We outline the proof and also remark our perspective on Idelic class field theory on a 3-manifold and Artin Lfunctions.
Next, we recall McMullen's examples and discuss the planetary links obtained from fibered hyperbolic links in S 3 . In addition, we recall an analogue of Artin's argument and exhibit the decomposition table in an analogue of a quintic field.
Finally, we recall the notions of (modulo 2) Olympic primes and links and calculate the density of modulo 2 Olympic links in a Chebotarev link.

CHEBOTAREV LINKS ARE STABLY GENERIC
Definitions and Theorems. We assume that any 3-manifold is the complement of a finite link in an oriented connected closed 3-manifold. A knot K in a 3-manifold M means a tame embedding S 1 = R/Z ãÑ M or its image with a natural orientation. A link is a countable (finite or infinite) set of disjoint knots. For any manifold X, we denote the interior of X by IntX.
McMullen [15] established a version of the Chebotarev density theorem in which number fields are replaced by 3-manifolds, answering to Mazur's question on the existence of Chebotarev arrangement of knots in [14]. Their definition is described as follows.
Definition 2.1 (Chebotarev law). Let (K i ) = (K i ) iPN ą0 be a sequence of disjoint knots in a 3-manifold M . For each n P N ą0 and j ą n, we put L n = Y iďn K n and denote the conjugacy class of K j in π 1 (M´L n ) by [K j ]. We say that (K i ) obeys the Chebotarev law if lim νÑ8 #tn ă j ď ν | ρ([K j ]) = Cu ν = #C #G holds for any n P N ą0 , any surjective homomorphism ρ : π 1 (M´L n ) Ñ G to any finite group, and any conjugacy class C Ă G. (The left hand side is the natural density of K i 's with ρ([K j ]) = C.) An infinite link K is said to be Chebotarev if it obeys the Chebotarev law with respect to some order.
On the other hand, Mihara [16] formulated an analogue of idelic class field theory for 3-manifolds by introducing certain infinite links called stably generic links, refining the notion of very admissible links given by Niibo and the author [20,21], and gave a cohomological interpretation of our previous formulation. Here we describe the definition of a stably generic link, only using ordinary terminology of low dimensional topology: Definition 2.2 (stably generic link). Let M be a 3-manifold and K ‰ ∅ a link. The link K is said to be generic if for any finite sublink L of K, the group H 1 (M´L) is generated by components of K´L. The link K is said to be stably generic if for any finite sublink L of K and for any finite branched cover h : M 1 Ñ M branched over L, the preimage h´1(K) is again a generic link of M 1 .
The following theorem is due to the author [32, Theorem 3]: Remark 2.4. For a sequence of knots ordered by length and obeying the Chebotarev law, we may define analogues of Artin L-functions (cf. [1,22,27,28]). In addition, Mihara's refinement allows us to study analogues of ray class fields. We expect that Theorem 2.3 would play a key role to expand an analogue of idèlic class field theory for 3-manifolds, in a direction of analytic number theory, with ample interesting examples. Another analogue in a more general setting are due to J. Kim (see ver.1 of [10]) and others [11].
Outline of the proof. A careful observation of the behavior of knots in a finite cover which is not necessarily Galois (regular) and the lifting property of a continuous map yield the following lemma. (1) Let K 1 be a connected component of h´1(K) in N . If the restriction map h| K 1 : K 1 Ñ K is a bijection, then the conjugacy classes satisfy We have an analogue of the Hilbert ramification theory for Galois branched covers of 3-manifolds, in which we describe the behavior of knots (instead of prime ideals) using the language of fundamental groups [31]. For a non-Galois cover h : N Ñ M and a knot K in M , the covering degrees of the restriction maps h| K 1 i : does not necessarily coincide with each other. In order to prove Theorem 2.3, we introduce the notion of a weakly Chebotarev link: Definition 2.6. Let M be a 3-manifold and K = Y iPN ą0 K i a countable link. The link K in M is said to be weakly Chebotarev if for any surjective homomorphism ρ : π 1 (M ) ↠ G to any finite group, any conjugacy class Lemma 2.5 yields the following key lemma.
We also have the following lemma.
Now Lemmas 2.7, 2.8, and a careful reading of the definition of a stably generic link yield Theorem 2.3.

EXAMPLES OF CHEBOTAREV LINKS
McMullen's results. McMullen proved that sequences of knots  (1) Let X be a closed surface of constant negative curvature, let M = T 1 (X) denote the unit tangent bundle, and let (K i ) denote the closed orbits of the geodesic flow in M , ordered by length.
(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. We consult [4,5] for terminology and basic facts related to pseudo-Anosov flows.
In [15], in order to connect symbolic dynamics to finite branched covers, McMullen proved an important lemma that assures that every conjugacy class of G is presented by a closed orbit, and invoked the notion of a Markov section. Then he applied a Chebotarev law for dynamical setting, which was proved by Parry-Pollicott [22,Theorem 8.5] with use of a method of Artin L-functions. We note that special cases of (1) and (2) were initially proved to obey the Chebotarev law by Adachi and Sunada in [27, Proposition II-2-12] and [1, Proposition C], the latter being related to topological entropy. (2) above is the planetary link K of a fibered hyperbolic finite link L in S 3 introduced by Birman and Williams [3], that is, the periodic orbits of the suspension flow of the monodromy map of L. By virtue of McMullen's theorem [15, Theorem 1.2] together with the Nilsen-Thurston uniformization theorem [30, Theorem 0.1], the union K Y L obeys the Chebotarev law, if ordered by length. We may prove in a similar way to Miller [17,Proposition 4.2] that for any finite sublink L 1 Ă K, the union L Y L 1 is again hyperbolic. Moreover, we may prove that such KYL is stably Chebotarev. See [32,Section 4].

Planetary links. An interesting example contained in
In addition to the Chebotarev law, the planetary link sometimes has another very noteworthy property. Ghrist and others proved that if a link L belongs to a certain large class of links containing the figure-eight knot, then the planetary link of L contains every links [6][7][8]. Hence we have a sequence (K i ) of knots containing every isotopy class of links and obeying the Chebotarev law. Moreover, as formulated by Kopei in [12], this example satisfies an analogue of the product formula |a| 8 ś p |a| p = 1 (a P Q), where p runs through all the prime numbers and |a| p denote the p-adic norm with |p| p = p´1. Therefore, the planetary link would give a fundamental setting, when we establish an analogue of number theory on 3-manifolds.

ARTIN'S ARGUMENT FOR A 5 -EXTENSIONS
Here we discuss an analogue of a quintic field. We first state the coincidence of the decomposition type of a knot and the cycle type of the monodromy permutation, which is an analogue of Artin's argument in [2] (see also [29,Chapter 16.2,Theorem 2]).

Definition 4.1.
(1) For each n P N ą0 , we denote the n-th symmetric group by S n and the n-th alternating group by A n . The cycle type of σ P S n is (f 1 ,¨¨¨, f r ) if it is the product of disjoint cycles of length f 1 ě¨¨¨ě f r .
(2) For a finite cover h : N Ñ M and a knot K Ă M , the decomposi- We may prove the following proposition in a parallel way to the one in number theory.  Let r h : W Ñ M denote the corresponding A 5 -cover. (Then the Fox completion W of W is a Poincaré 3-sphere, cf. [25].) Let H ă Gal(h) -A 5 be any subgroup of index 5, and let h : N Ñ M denote the corresponding subcover of degree 5. The kernel Ker(ρ) of the monodromy permutation ρ : A 5 Ñ S 5 coincides with the normalizer of H in A 5 . Since A 5 is a simple group, Ker(ρ) = tidu holds and ρ is an injection. By Proposition 4.2 together with the Chebotarev law applied for the composite the number of elements of A 5 of each cycle type and the natural density of K i 's of each decomposition type are given as follows.
Artin L-functions of symbolic flows due to Parry-Pollicott [22] are regarded as that of Chebotarev links. We may also discuss an analogue of Artin's argument in [2] with use of L-functions associated to the setting in Example 4.3.

OLYMPIC PRIMES AND LINKS
The analogy between the Legendre symbol ( p q ) for two prime numbers p, q and the modulo 2 linking number lk 2 (K, L) of a two-component link K \ L is the most classical one in the dictionary of arithmetic topology. Some people say that it was initially noticed by Gauss, and indeed we may find some ideas in his proofs of the quadratic reciprocity 2 q which might have come from his study in electromagnetism.
An Olympic prime is a 5-tuple (p 1 , p 2 , p 3 , p 4 , p 5 ) of prime numbers satisfying An example is displayed in the figure above. On the first day of Morishita's course in arithmetic topology for undergrads in October 2012, he asked to prove that there are infinitely many Olympic primes. By using Dirichlet's Theorem [26, Chapter VI, Theorem 2], for instance, we may easily calculate that the natural density of Olympic primes is d = 1 2 10 , where 10 = Similarly, a modulo 2 Olympic link is a link L 1 Y¨¨¨Y L 5 with 5 components in S 3 satisfying lk 2 (L i , L i+1 ) = 1 for i = 1, 2, 3, 4 and lk 2 (L i , L j ) = 0 for i + 1 ă j. Now let (K i ) iPN ą0 be a sequence of disjoint knots in S 3 obeying the Chebotarev law and put K = Y The similar argument holds for modulo m linking number for every positive integer m. However, this does not tell whether there exists an Olympic link without modulo.
It might be also interesting to investigate the density of 3-component links with given Milnor's triple linking number. ACKNOWLEDGMENTS I would like to express my sincere gratitude to Sergey Maksymenko for inviting me to the workshop "Morse theory and its applications" in Kyiv, Ukraine. I was much inspired and encouraged by the atmosphere and culture there. I am also grateful to people who I met in Göttingen, Germany. When I visited the Gaußhaus, I felt that it is natural to expect his intuition of electromagnetism among prime numbers. Finally I would like to thank Hirofumi Niibo for asking an important question to clarify the argument in this manuscript, and the guy who gave me a copy of the lecture note containing the handwriting version of the figure above.