TITLE: Is there a periodic table for knots?
QUESTION [15 upvotes]: When I see knot tables, I have two feeling: ah, it's beautiful, and... painful.
I don't see how knots are ordered in the knot table, the way to go from one knot of a certain crossing number to another seems to be completely random. But I would guess there are some order? For example, why are the Perko pair put next to each other even before people knew they are the same?
In short, if the word "periodic table" seems confusing, my real question is, how are the knots in knot table ordered?

REPLY [3 votes]: I'm surprised no one mentioned the story by William Thompson, also known as lord Kelvin, which originally thought of atoms or elements as knots! That would literally put them in a periodic table :).
I learned this from  this fascinating talk by Haynes Miller, which I quote the relevant part

Thomson’s vision was that perhaps atoms consisted
of tiny vortex rings, persistent topological singularities in the aether. Different elements
might correspond to different knot types!—so perhaps the hydrogen atom was a simple
loop (the “unknot”), helium was the trefoil knot, . . . .  These rings vibrated, putting them in
different energy states. And perhaps the mystery of the formation of molecules could be
explained by the linking of several knots to form links.