[Math] periodic table for knots

Question

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?

Answer 1

Just for kicks, here's a partial list of various ways some people like to occasionally think of as ways of sorting knots.

• Knot energies. For example, the electrostatic potential on knots in $S^3$ is a real-valued function on the space of knots in $S^3$ such that there's only finitely-many knot types below any given energy level. See papers of Freedman, He and Wang, like Möbius invariance of knot energy, also Jun O'Hara. But there are many other knot energies out there in the literature.

• Crossing number + ??. The traditional knot table. Closely related are things like bridge numbers. Minimal number of tetrahedra in a triangulation of the complement. Stick number. Degree of a polynomial or trig function that it takes to represent the knot, and so on.

• Geometrization (as I mentioned in my comments above). See also Daniel's comment.

• Geometrization + the geometrization of the 2-sheeted cyclic branched cover of $(S^3,K)$. This is related to "arborescent knots". Similarly, this leads to all kinds of variant ideas. See the big paper of Bonahon and Siebenmann. This is also related to rational tangle decompositions of knots.

• Braid index + a canonical form for conjugacy classes in the braid group.

• Plat closures + canonical representatives of double-cosets of the Hilden / wicket subgroup. This would be a refinement of the bridge number description.

• You could sort knots based on various knot invariants. Alexander polynomials and Jones polynomials being fairly popular ones.

edit: Ken Perko wrote to me to object to my first comment (top of the page, before my answer). His comment deserves a post of his own but until that happens, I'll quote him here:

I beg to disagree with your comment that it's just based on the order in which they were discovered -- except, of course, for increasing crossing numbers tabulated by different people at different times.

Tait and Little seem to have organized the order within a given crossing number by their own criteria of how the knots looked to them -- Little famously using, in his non-alternating 10-crossing list, the so-called invariant of "twist" (now known as writhe) which placed the two copies of the Perko pair knots far apart from each other. Alexander and Briggs looked to 2-fold homology (which makes a lot of sense and was copied by Reidemeister) and Rolfsen used the Alexander polynomial, which for the first time put the Perko pair knots next to each other (not that that helps very much in seeing that they are the same). I wouldn't know how to describe the order established by Conway, Thistlethwaite and the rest for non-alternating 11's and 12's and on up, but I don't think the order of discovery had much to do with it.Conway followed his own peculiar patterns and Thistlethwaite and successors may have just left it all up to the machines,

Nonetheless, your analysis is quite correct for the four knots added to Conway's published table and shown at the end of page 117 of Topology Proceedings 7 (1982). The first two were listed in D. Lombardero's 1968 Princeton senior thesis (of which one is the likely explanation for a typographical duplicate in Conway's paper) and the last two were discovered in the late 1970's by A. Caudron.