I haven’t blogged for a long time- for the last few years, my research has been leading me away from low dimensional topology and more towards foundations of quantum physics. You can read my latest paper on the topic HERE.

Today I’d like to tell you about a preprint by Malyutin, that shows that two widely believed knot theory conjectures are mutually exclusive!

A Malyutin, On the Question of Genericity of Hyperbolic Knots, https://arxiv.org/abs/1612.03368.

Conjecture 1: Almost all prime knots are hyperbolic. More precisely, the proportion of hyperbolic knots amongst all prime knots of or fewer crossings approaches 1 as approaches .

Conjecture 2: 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.

Both conjectures have a lot of numerical evidence to support them.

For conjecture 1, just look at the following table, cited by Malyutin from Sloane’s encyclopedia of integer sequences:

Also, many topological objects that are related to knots are generically hyperbolic- e.g. compact surfaces, various classes of 3-manifolds, closures of random braids…

For conjecture 2, two stronger conjectures are widely believed. First, that crossing number should in fact be additive with respect to connect sum (so the crossing number of a composite knot should in fact be the sum of the crossing number of its components). This has been proven for various classes of knots- alternating knots, adequate knots, torus knots, etc. Secondly, that the crossing number of a satellite knot is not less than that of its companion.

Which of these seemingly-obvious, and widely-believed, conjectures is false?? This is high drama in the making!