Knots which behave like the prime numbers
This article is a contribution to the growing amount of results that show the existence of a close analogy between the theory of knots and number theory see \textit{M. Morishita} [Knots and primes. An introduction to arithmetic topology. (Springer, 2009); Universitext. Berlin: Springer (2012; Zbl 1267.57001)]. In the present case, the author establishes a version of the Chebotarev density theorem for \(3\)-manifolds.NEWLINENEWLINEIn fact, let \((K_j)\) be a sequence of disjoint, smooth, oriented knots in a closed, connected \(3\)-manifold \(M\), set \(L_n = \bigcup_1^{n} K_n\), and let \(G\) be a finite group. Any surjective homomorphism \(\rho: \pi_1(M \setminus L_n) \longrightarrow G\) determines a covering space \(\widetilde{M} \longrightarrow M\) with Galois group \(K\). The unramified knots \(K_{n+1}\), \(K_{n+2}, \ldots\) determine a sequence of conjugacy classes \([K_j]\) in \(G\).NEWLINENEWLINEAccording to Mazur, the sequence of knots is said to obey the Chebotarev law if for any \(\rho\) as above and any conjugacy class \(C\) in \(G\) we have NEWLINE\[NEWLINE \lim_{N \to \infty} \frac{|\{ n < i \leq N: [K_i] = C\}|}{N} = \frac{|C|}{|G|}\;.NEWLINE\]NEWLINE The main result of this article is the following theorem: Let \(X\) be a closed surface of constant negative curvature, and let \(K_1, K_2, \ldots \subset M = T_1(X)\) be the closed orbits of the geodesic flow, ordered by length. Then \((K_i)\) obeys the Chebotarev law.
scientific article; zbMATH DE number 6213918
scientific article