Chebotarev links are stably generic

Abstract

We discuss the relationship between two analogues in a 3‐manifold of the set of prime ideals in a number field. We prove that if false(Kifalse)i∈N>0 is a sequence of knots obeying the Chebotarev law in the sense of Mazur and McMullen, then K=∪iKi is a stably generic link in the sense of Mihara. An example we investigate is the planetary link of a fibered hyperbolic finite link in S3. We also observe a Chebotarev phenomenon of knot decomposition in a degree 5 non‐Galois subcover of an A5(icosahedral)‐cover.

Twisted Iwasawa invariants of knots

“…[62, Remark 4.2]), so are the Iwasawa invari-ants, noting that our Iwasawa invariants are defined if and only if the Iwasawa module  𝜌,𝑖 is a torsion 𝑂 𝔭 [[𝑡 ℤ 𝑝 ]]-module. Hence, we obtain the following result on the profinite rigidity, refining [4, Theorem 1.2] and [64]. Proof.…”

“…[62, Remark 4.2]), so are the Iwasawa invariants, noting that our Iwasawa invariants are defined if and only if the Iwasawa module $${\mathcal {A}}_{\rho ,i}$$ is a torsion $$O_{\mathfrak {p}}[[t^{{\mathbb {Z}}_{p}}]]$$‐module. Hence, we obtain the following result on the profinite rigidity, refining [4, Theorem 1.2] and [64]. Theorem The fiberedness and the genus of a knot $$K$$ are determined by the isomorphism class of the profinite completion of the knot group $$\pi$$, via the twisted Iwasawa invariants.…”

Modular knots, automorphic forms, and the Rademacher symbols for triangle groups

“…The set of modular knots around the trefoil satisfies another distribution formula called the Chebotarev law in the sense of Mazur [ 25 ] and McMullen [ 26 ], so that it may be seen as an analogue of the set of all prime numbers in [ 43 , 44 ], in a sense of arithmetic topology [ 29 ]. An exploration of a unified viewpoint for these formulas would be of further interest.…”

Olympic links in a Chebotarev link

“…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.…”