On the question of genericity of hyperbolic knots. (English) Zbl 1473.57017
A well-known conjecture in knot theory states:
Conjecture 1. The proportion of hyperbolic knots among all of the prime knots of \(n\) or fewer crossings approaches 1 as \(n\) approaches infinity.
Current knot tables (up to 19 crossings) show that the overwhelming majority of prime knots with a small crossing number is indeed hyperbolic. In this article, the author proves that Conjecture 1 contradicts several other plausible conjectures, such as the following:
Conjecture 2. The crossing number of knots is additive with respect to connected sum.
Conjecture 3. The crossing number of a satellite knot is not less than that of its companion.
Conjecture 4. The crossing number of a composite knot is not less than that of each of its factors.
If \(P\) is a knot and \(\lambda\) is a real number, we say that \(P\) is \(\lambda\)-regular if we have \(cr(K) \ge \lambda cr(P)\) whenever \(P\) is a factor of a knot \(K\).
Conjecture 5. Each knot is \(\frac{2}{3}\)-regular.
Conjecture 6. There exist \(\epsilon > 0\) and \(N > 0\) such that, for all \(n > N\), the proportion of \(\frac{2}{3}\)-regular knots among all of the hyperbolic knots of \(n\) or fewer crossings is at least \(\epsilon\).
The main theorem of the article is the following:
Conjecture 1 contradicts Conjecture 6 and hence it also contradicts (each of the) Conjectures 2, 3, 4, and 5.
Conjecture 1. The proportion of hyperbolic knots among all of the prime knots of \(n\) or fewer crossings approaches 1 as \(n\) approaches infinity.
Current knot tables (up to 19 crossings) show that the overwhelming majority of prime knots with a small crossing number is indeed hyperbolic. In this article, the author proves that Conjecture 1 contradicts several other plausible conjectures, such as the following:
Conjecture 2. The crossing number of knots is additive with respect to connected sum.
Conjecture 3. The crossing number of a satellite knot is not less than that of its companion.
Conjecture 4. The crossing number of a composite knot is not less than that of each of its factors.
If \(P\) is a knot and \(\lambda\) is a real number, we say that \(P\) is \(\lambda\)-regular if we have \(cr(K) \ge \lambda cr(P)\) whenever \(P\) is a factor of a knot \(K\).
Conjecture 5. Each knot is \(\frac{2}{3}\)-regular.
Conjecture 6. There exist \(\epsilon > 0\) and \(N > 0\) such that, for all \(n > N\), the proportion of \(\frac{2}{3}\)-regular knots among all of the hyperbolic knots of \(n\) or fewer crossings is at least \(\epsilon\).
The main theorem of the article is the following:
Conjecture 1 contradicts Conjecture 6 and hence it also contradicts (each of the) Conjectures 2, 3, 4, and 5.
Reviewer: Claus Ernst (Bowling Green)
MSC:
57K10 | Knot theory |