Knotted

Why lighting it proves it

Wiggle a knotted loop of rope however you like, without cutting it, and its picture on the table changes. The picture can get as messy as you please, so failing to untie it proves nothing, however long you try: the next wiggle might be the one that undoes it. (There is a limit: Marc Lackenby proved in 2015 that a picture of an untied loop with c crossings can always be undone in at most (236c)¹¹ moves, so trying every way of making that many moves would settle it, but for a picture with only 3 crossings that is more than 10³¹ moves.) A practical proof finds something the wiggling cannot change.

So count the lightings of a picture, because no wiggle can change the count. Every wiggle, seen in the picture, comes down to three small moves (plus smooth stretching that leaves the crossings alone): put a twist in a strand or take one out, slide one strand across another, slide a strand across a crossing. Two pictures show the same knot exactly when a chain of these moves turns one into the other. Kurt Reidemeister proved it in a paper written in January 1926, and J. W. Alexander and his student G. B. Briggs proved it too, in a paper received in April 1927 that cites his. Try each move on a lit picture and the lighting carries across, one for one, so the number of lightings belongs to the knot and not to the picture.

Above, the trefoil as the knot table draws it has 9 lightings with three colours, and the same knot wiggled into 11 crossings still has 9. An untied loop can be drawn as a plain circle, a single strand, which takes one colour at a time: with p colours it has p lightings, each in a single colour, however it is tangled. The loop above has 9 crossings and 3 lightings. The knot you lit has a lighting that uses more than one colour. No wiggle can make that go away, so no wiggle can untie it.

The two wiggled pictures were made by random Reidemeister moves, applied by spherogram (part of SnapPy) from a fixed seed, and spherogram’s own simplifier takes them back down: the trefoil to its 3 crossings, the loop to none.

Why a mirror

Knot theorists call a lighting a Fox colouring, after Ralph Fox. Józef Przytycki, who has written its history, thinks Fox devised the three-colour version around 1956, while explaining knot theory to undergraduates at Haverford College, though he notes “some controversy who really invented them”. Number the colours round the wheel from 0, and Fox’s rule at a crossing is one line of arithmetic: twice the number on top equals the sum of the two underneath, counting round the wheel (on a wheel of 5, 4 + 3 counts as 2).

On a wheel with an odd number of colours, that arithmetic is a reflection. Draw the colours as the corners of a regular polygon, and the two colours underneath are mirror images of each other across the line through the corner on top. With three colours the polygon is a triangle. Reflecting in one corner leaves that corner where it is and swaps the other two, so the two strands underneath are either both the colour on top or the other two colours: all the same, or all different, which is the three-crayon rule. The wheel beside the knot draws the mirror for whichever colour you are holding.

The mirrors are meant literally. Przytycki gives the exact statement: the colourings of a knot with n colours correspond one for one to the ways of mapping the knot’s group to the symmetries of a regular n-sided polygon that send each strand to a reflection. He also notes that Reidemeister had already studied such maps, which lead straight to Fox’s colourings: one reason the question of who invented them is not settled.

Which wheels can light which knots

Not every wheel can light every knot with more than one colour. For a wheel whose number of colours is prime, one number decides it: the knot's determinant. A wheel works exactly when its size divides the determinant. On a wheel whose size shares no factor with the determinant, the only lightings are the one-colour ones.

The page works the determinant out from the drawing in front of you: write the rule as one equation per crossing, strike out one equation and one strand, and take the determinant of what is left, ignoring its sign (exact integer arithmetic, done in your browser). Open Show the working below the puzzle to see it for the knot you are playing, beside the value the knot table publishes.

The knots no wheel can light

Every wheel fails, and 10₁₂₄ is knotted all the same: it is the torus knot T(5,3), a curve drawn on the surface of a ring that wraps five times around it one way and three times the other. A lighting that uses more than one colour proves a knot is knotted. Not finding one proves nothing, because some knots have none. Among the seventeen are two famous knots, the Kinoshita–Terasaka knot and the Conway knot (11n₄₂ and 11n₃₄ in the table, each up to its mirror image), whose Alexander polynomials are 1.

A sharper instrument does see 10₁₂₄. The Jones polynomial, which Vaughan Jones published in 1985, computed here from the same drawing by Louis Kauffman’s state sum, is t⁴ + t⁶ − t¹⁰ for 10₁₂₄, and it is 1 for every untied loop. Whether some knotted loop also has Jones polynomial 1 is the question Jones put to the twenty-first century. A 2024 paper on it calls the answer “still unknown despite numerous attempts to explore it”. In a paper published in 2021, Robert Tuzun and Adam Sikora report checking every knotted loop with a diagram of up to 24 crossings by computer: none has. A preprint posted in June 2026 by Avishy Carmi and Eliahu Cohen claims a proof that none ever does; it lists no journal, and when we looked (8 November 2026) we found no published check of it by anyone else, so on this page the question stays open. A finer instrument built on the Jones polynomial, Khovanov homology, does tell every knot from an untied loop, as Peter Kronheimer and Tomasz Mrowka proved in a paper published in 2011.

Where the knots come from

Every knot here is a prime knot from KnotInfo, the table of knot invariants kept by Charles Livingston and Allison H. Moore: all 2,977 with 3 to 12 crossings, as shipped in version 2026.11.2 of the database_knotinfo package. Each drawing is made from KnotInfo’s own record of the knot’s crossings (its PD code), laid out on a grid by the orthogonal layout in spherogram, part of SnapPy, which follows Roberto Tamassia’s 1987 method for drawing a graph on a grid with as few bends as possible. The layout is not taken on trust. Before the data was written, the engine this page runs re-read every drawing as a knot diagram and held it up against KnotInfo: it is KnotInfo’s diagram (the same crossings, joined the same way, the same strand on top at each, up to mirror image), and it has KnotInfo’s determinant. The check below adds a third comparison: each drawing has exactly the Jones polynomial KnotInfo lists. 2,977 of 2,977 pass all three. The comparisons can fail, and do when they should: swap the strands at one crossing of each drawing and every one of the 2,977 stops matching; hold each drawing up against the next knot in the table and none of the 2,967 pairs matches.

How a daily is made

Each puzzle starts from a lighting that proves its knot, picked at random from all of them, preferring ones that use more of the wheel. Then, in a random order, each strand’s colour is taken away if single-crossing deductions can still put it back. What is left are the given strands. From them every other strand is forced, one crossing at a time, so there is one answer and no guessing, and an elimination over the whole wheel confirms that no other lighting fits the givens at all. Every given is needed: take any one away and the deductions stall. All 730 dailies, one a day from 9 November 2026, were made and checked this way before they were scheduled, and Endless makes its levels the same way in your browser.

Check it yourself

Everything above about the drawings and the puzzles, apart from what spherogram did, is re-derived by one program, check.mjs, served beside this page. Save it anywhere and run node check.mjs with Node 18 or later: it downloads the engine, the drawings, the dailies, the two wiggled pictures and the copy of KnotInfo from this site and runs 32 checks, controls included, in about a minute (most of it the Jones polynomials). The copy of KnotInfo it compares against is knotinfo.json; any row of it can be compared with KnotInfo’s own site. What spherogram did can be redone with spherogram (pip install spherogram): Link(pd).simplify('global') on the PD codes in wiggles.json brings the wiggled trefoil back to 3 crossings and the loop to none.

Sources: C. Livingston and A. H. Moore, KnotInfo: Table of Knot Invariants, knotinfo.org, November 2026, read through the database_knotinfo package, version 2026.11.2. M. Culler, N. M. Dunfield, M. Goerner and J. R. Weeks, SnapPy, a computer program for studying the geometry and topology of 3-manifolds, snappy.computop.org (spherogram 2.4.1). R. Tamassia, “On embedding a graph in the grid with the minimum number of bends”, SIAM J. Comput. 16 (1987) 421–444. K. Reidemeister, “Elementare Begründung der Knotentheorie”, Abh. Math. Sem. Univ. Hamburg 5 (1926) 24–32; J. W. Alexander and G. B. Briggs, “On types of knotted curves”, Ann. of Math. 28 (1926–27) 562–586; both as recounted by H. Queffelec, “Reidemeister’s theorem using transversality”, arXiv:2406.18203. M. Lackenby, “A polynomial upper bound on Reidemeister moves”, Ann. of Math. 182 (2015) 491–564, arXiv:1302.0180. J. H. Przytycki, “3-coloring and other elementary invariants of knots”, Banach Center Publ. 42 (1998), arXiv:math/0608172 (Lemma 3.10); and “Knot Theory: from Fox 3-colorings of links to Yang-Baxter homology and Khovanov homology”, arXiv:1809.00063. P. Lopes and J. Matias, “Minimum number of Fox colors for small primes”, J. Knot Theory Ramifications 21 (2012) 1250025, arXiv:1001.1334 (Section 2.1). L. H. Kauffman and P. Lopes, “Determinants of rational knots”, Discrete Math. Theor. Comput. Sci. 11 (2009), arXiv:0710.3765. V. F. R. Jones, “A polynomial invariant for knots via von Neumann algebras”, Bull. Amer. Math. Soc. 12 (1985) 103–111. V. F. R. Jones, “Ten problems”, in Mathematics: Frontiers and Perspectives (AMS, 2000) 79–91. L. H. Kauffman, “State models and the Jones polynomial”, Topology 26 (1987) 395–407. D. Korzun, E. Lanina and A. Sleptsov, “Closed 4-braids and the Jones unknot conjecture”, Ann. Inst. Henri Poincaré D 13 (2025) 711–746, arXiv:2402.02553. R. E. Tuzun and A. S. Sikora, “Verification of the Jones unknot conjecture up to 24 crossings”, J. Knot Theory Ramifications 30 (2021) 2150020, arXiv:2003.06724. A. Carmi and E. Cohen, “On trivial Jones–Vassiliev polynomials”, arXiv:2606.22410 (preprint, June 2026). M. Khovanov, “A categorification of the Jones polynomial”, Duke Math. J. 101 (2000) 359–426, arXiv:math/9908171. P. B. Kronheimer and T. S. Mrowka, “Khovanov homology is an unknot-detector”, Publ. Math. IHÉS 113 (2011) 97–208, arXiv:1005.4346. The Knot Atlas pages for 10_124, K11n34 and K11n42 (katlas.org).