Two widely believed conjectures:
- The proportion of hyperbolic knots amongst prime knots of $n$ or fewer crossings approaches $1$ as $n$ approaches infinity.
- The crossing number (the minimal number of crossings of a knot diagram of that knot) of a composite knot is not less than that of each of its factors.
The first conjecture is widely believed because of massive numerical evidence, and because many topological objects that are related to knots are generically hyperbolic- e.g. compact surfaces, various classes of 3-manifolds, closures of random braids...
The second conjecture is widely believed to be true because of lots of numerical evidence- it is moreover believed that crossing number should in fact be additive with respect to connect sum (so the crossing number of a composite knot should actually be the sum of the crossing number of its components). It was proved for various classes of knots- alternating, adequate, torus, etc. It is implied by the much stronger Petronio-Zanellati Conjecture which also has strong numerical evidence supporting it.
... and yet ...Malyutin shows that either Conjecture 1 or Conjecture 2 must be false!!