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.