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 
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!
Low Dimensional Topology 
I would assume the Adams conjecture (Conjecture 1) is false. While prime satellite knots aren’t a big proportion of all prime knots (counting via crossings in planar diagrams) I don’t see how they could be vanishingly small. Asymptotically I would imagine they are some small non-zero percentage of all prime knots.
Comment by Ryan Budney — April 4, 2018 @ 12:19 pm |
My feeling is Conjecture 1 is very likely false, even without reference to its relationship to Conjecture 2. In fact, I would go even further than Ryan and conjecture that prime knots are satellites with probability tending to 1 as the crossing number goes to infinity. Here are two papers that show closely related statements, the difference being that they are looking at random *knot diagrams* with n crossings, not random *knots* whose minimal projection have n crossings. For example, the second one discusses a model of random knot diagram and proves (Corollary 1.3) that the expected number of components of the JSJ decomposition grows linearly in n. My intuition is that the distinction between diagrams and knots will make no difference, not that I have any idea how one would prove this…
https://arxiv.org/abs/1608.02638, https://arxiv.org/abs/1611.04944
Comment by Nathan Dunfield — April 4, 2018 @ 2:48 pm |
BTW, the reason that this doesn’t contradict e.g. the fact that the closure of a random braid is hyperbolic is that these are very different models of random knot: random diagrams have “local knotting” and random braid closures do not. See here for a nice survey: https://arxiv.org/abs/1711.10470
Comment by Nathan Dunfield — April 4, 2018 @ 2:53 pm |
I also believe that Conjecture 1 is false. The argument in the paper uses the fact that the number of diagrams with n crossing grows “only” exponentially in n, so if from a hyperbolic knot with n crossing you can construct a satellite one with n + 100 crossing (and you prove that distinct knots give distinct satellite) then you disprove Conjecture 1. This argument however does not work if one considers triangulations since they grow more than exponentially fast. Both the number of hyperbolic and non-hyperbolic manifolds triangulated with n tetrahedra grow faster than exponentially and it is hard to tell who is going to win the race…
Comment by Bruno Martelli — April 4, 2018 @ 4:22 pm |
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Pingback by On the Low-Dimensional Topology blog: “two widely believed knot theory conjectures are mutually exclusive” – Nevin Manimala’s Blog — April 5, 2018 @ 4:14 pm |
Presumably Lackenby’s work is enough to disprove the Adams Conjecture. http://people.maths.ox.ac.uk/lackenby/csa20813.pdf See the two theorems he lists in the introduction to the paper.
Comment by Ryan Budney — May 19, 2019 @ 7:38 am |