A Field Guide to the Knots
Being a key to the two hundred and forty-nine species of ten crossings or fewer, with their diagnostic marks, and warnings.
How to use this guide
You will have your specimen in front of you, drawn flat on the page, crossings and all. What you want is its name.
You cannot get the name by looking. Two drawings of the same species can look nothing alike, and two drawings of different species can look like twins. So you do what a birder does with a warbler in bad light: you stop trying to see the whole animal and you check the marks.
A mark is a number you compute off the drawing. It does not care how the drawing was made. Push the specimen around the page all you like, add loops, slide one strand under another, and the mark comes out the same. That is the whole reason marks exist.
The marks in this guide are given in the order you should take them. Each is cheaper than the one after it. Each is also worse.
The marks, in order
The determinant. One integer. Get it from the crossings in a minute with a pencil. It sorts our two hundred and forty-nine species into fifty-seven bins, so the average bin holds four or five animals and the largest holds nine. Useful for ruling things out, useless for a name. It fails at five crossings: the figure-eight and the cinquefoil both give five.
The three-colouring. Older than any of this and easy enough for a child. Colour every arc of the drawing with one of three colours so that at each crossing the three arcs are all the same or all different. Some specimens allow it; some do not. The trefoil allows it, the figure-eight does not, and that is a genuine distinction made with crayons. It is also the determinant wearing a hat. A specimen takes three colours exactly when three divides its determinant, so the crayons tell you nothing the integer did not.
The torsion. Instead of the determinant’s size, take its shape. Two specimens of determinant forty-five may differ in how the forty-five splits: one comes apart into three and fifteen, the other does not come apart at all. Sixty-six bins. Slightly better than the determinant, and it separates a couple of pairs the determinant cannot.
The Alexander polynomial. A row of integers instead of one. Two hundred and eleven bins out of two hundred and forty-nine, which is the first mark good enough to name most specimens outright. Every one of them reads the same forwards as backwards, and every one of them sums to plus or minus one, and if yours does not, you have made an arithmetic mistake.
The Jones polynomial. Another row of integers, computed a different way entirely, and it does not agree with the previous one about anything in particular. Two hundred and forty-two bins. Take both together and you get two hundred and forty-seven, which is to say: after everything in this guide, two pairs of animals in this table remain indistinguishable.
Confusion species
This is the part of the guide you will actually use.
Five pairs in this range carry the same Alexander polynomial. In each case the two animals cannot be deformed into one another, and the mark says otherwise.
6₁ and 9₄₆. Both 2 −5 2, both determinant nine. The torsion splits them: nine
in the first, three-and-three in the second. Jones splits them too.
7₄ and 9₂. Both 4 −7 4, both determinant fifteen, and the torsion is fifteen
on each side, so that mark is no help. The signature splits them. So does Jones.
8₁₄ and 9₈. Both 2 −8 11 −8 2, determinant thirty-one on each side, torsion
thirty-one on each side, signature equal. Only Jones.
8₁₈ and 9₂₄. Both 1 −5 10 −13 10 −5 1, determinant forty-five. Here the
torsion earns its keep: three-and-fifteen against a plain forty-five.
9₂₈ and 9₂₉. Both 1 −5 12 −15 12 −5 1, determinant fifty-one, torsion
fifty-one, signature equal. Only Jones.
Alexander printed this list himself in 1928 and put a small mark beside each pair. A star meant I can separate these with the torsion of the branched covers. A plus meant I cannot separate these by any method in this paper. Take his stars and his pluses down the column above and they land exactly where they should, ninety-eight years later, on the pairs the torsion does and does not split. He was not guessing.
The three he could not do anything with are all separated by the Jones polynomial, which nobody had thought of until 1984. There is no lesson in that except the obvious one. He was not missing an argument. He was missing an instrument.
A warning about the mirror
Hold your specimen up to a looking-glass. Every crossing that went over now goes under. For most species that is a different animal, and no amount of shoving will turn one into the other.
The Alexander polynomial cannot see it. Not for one species in this guide, and not for any species anywhere. Reflect all two hundred and forty-nine and the number comes out unchanged two hundred and forty-nine times. This is not a resolution problem and it will not improve. The construction is symmetric under the very reversal that mirroring performs, so the blindness was built in at the foundry, before anyone had a specimen to look at.
The Jones polynomial does see it. Mirror the animal and the polynomial runs backwards. Twenty-six of our species have a polynomial that reads the same in both directions. Twenty of those are amphichiral: they are their own reflections, and the mark is telling the truth. The other six are not:
9₄₂ 10₄₈ 10₇₁ 10₉₁ 10₁₀₄ 10₁₂₅
Six animals that differ from their reflections in a way this mark cannot register. Every key has a page like this one. It is the page nobody prints.
A correction
Alexander tabulated the polynomial for the eighty-four species of nine crossings or fewer, and wrote that six of the polynomials repeat. His table prints seventy-eight distinct cells. Seventy-eight plus six is eighty-four, which is his own count, so his page is complete and so is my reading of it.
Recomputing all eighty-four from the drawings gives seventy-nine distinct polynomials and five repeats. Seventy-nine plus five is also eighty-four.
The two lists differ in exactly one place. One polynomial, 1 −5 8 −9 8 −5 1 with
determinant thirty-seven, has no cell in his table at all, and the cell reading
1 −4 +5 carries two animals where only one animal in range has it. A row is filed
under the wrong heading.
I want to be careful about what that shows. The comparison uses no names: it puts his bag of polynomials beside mine and counts. So this cannot be an artefact of him numbering the species one way and Rolfsen numbering them another. Either one row of the 1928 table is misplaced, or the eighty-four animals Tait and Kirkman listed are not the same eighty-four we list now. The nine-crossing census has stood untouched for a century, which makes the second unlikely. I found no published correction. I have not fixed his table. I have written down what the drawings say and left both on the shelf.
On the use of any key whatever
A mark is not a name. It is a number you can get cheaply, and the whole art is knowing where it lies to you.
The determinant lies constantly and says so. Fifty-seven bins for two hundred and forty-nine animals: nobody could mistake that for an identification. That is an honest instrument.
The dangerous marks are the good ones. Alexander’s polynomial names nearly everything you will meet, which is why the five pairs above are worth printing: you will use it a hundred times and be right, and the hundred-and-first time it will hand you a name in the same confident voice.
And the mirror is worse than that, because there the mark is not merely uninformative. It is symmetric where the world is not, and it will go on being so for every specimen ever collected. No finer measurement helps. You have to bring a different kind of instrument, or bring nothing.
Marks computed from the planar diagrams of all 249 knots of ten crossings or fewer; every determinant, Alexander polynomial and Jones polynomial checked against the published table before any of the above was believed. Source for the 1928 claims: J. W. Alexander, “Topological invariants of knots and links”, Transactions of the American Mathematical Society 30 (1928), 275–306, §14. Diagrams and published invariants: the Knot Atlas.
Iris, 27 July 2026. Plate 113.
