Hyperbolic links are not generic

Abstract

We show that if K is a nontrivial knot then the proportion of satellites of K among all of the prime non-split links of n or fewer crossings does not converge to 0 as n approaches infinity. This implies in particular that the proportion of hyperbolic links among all of the prime non-split links of n or fewer crossings does not converge to 1 as n approaches infinity. We consider unoriented link types.

Hyperbolic knots are not generic

“…We conjecture that for any nontrivial knot K the proportion of satellites of K among all of the prime knots of n or fewer crossings tends to 1 as n approaches infinity. Certain modifications of techniques developed in [Mal19] significantly strengthen the estimates of Theorems 1 and 2, however, as far as we know, these techniques are not enough to prove this conjecture.…”

“…In order to prove Theorem 2, it is enough to replace "prime non-split links" with "prime knots" in the proof of Theorem 1 in [Mal19] and observe that, by Corollary 1, assertion (iv) of Proposition 1 in [Mal19] holds for all prime knots.…”

“…In particular, we kindly refer the reader to these two papers and references therein for the terminology and introduction to the subject. In this note, we proceed directly with a new lemma (see Lemma 1 below) we just discovered and explain how to plug this lemma into constructions of [Mal18] and [Mal19] in order to prove that the proportion of hyperbolic knots among all of the prime knots of n or fewer crossings does not converge to 1 as n approaches infinity (this disproves a well-known conjecture; see [Ad94,p. 119]) and, moreover, that for any nontrivial knot K the proportion of satellites of K among all of the prime knots of n or fewer crossings does not converge to 0 as n approaches infinity.…”

“…The present note is intended as an extension and composed as an addendum to [Mal18] and [Mal19], where the question of genericity of hyperbolic knots and links is studied. In particular, we kindly refer the reader to these two papers and references therein for the terminology and introduction to the subject.…”

“…A brief explanation of the subject matter is as follows. One of the key ingredients of arguments in [Mal18] and [Mal19] are specific ways of constructing satellite knots with local satellite structures (γ-knots and K-entanglements) from nonsatellite ones. A difficult point there is showing that the obtained satellites are prime.…”