A tangle that is a fraction

John Conway liked to demonstrate a theorem with four volunteers and two ropes. Each volunteer holds one end, standing at the corners of a square. Two moves are allowed. In a twist, the two people on the right swap places, the one in front passing the other, which wraps the ropes once more round each other. In a rotation, all four move one place round the square, a quarter-turn. After any sequence of these moves the ropes are tangled, and Conway would ask the audience to untangle them using only the same two moves — never undoing a twist directly.

The trick is that every tangle reachable this way carries a fraction, the moves change it in a simple way, and the fraction says exactly how tangled the ropes are. Conway proved in 1970 that two of these rational tangles can be deformed into each other exactly when their fractions are equal. Joining the ends of a rational tangle in pairs makes a knot or a link, and the fraction then names the knot, by a rule Horst Schubert had found in 1956. This essay draws the knots from their fractions, counts them, and follows the dance on the Farey graph.

Two moves and their arithmetic #

Start with two parallel ropes, the tangle with fraction 0. A twist adds a half-turn to the right-hand end, and Conway’s rule is that it adds 1 to the fraction: $x \mapsto x + 1$. A rotation turns the whole tangle through a quarter-turn and reflects it in a mirror, and it sends $x \mapsto -1/x$. Three twists from 0 give the tangle 3, three half-twists side by side. A rotation of that gives $-1/3$, the same three twists standing vertically. Every rational tangle is a sequence of these moves applied to 0, and its fraction is computed by applying the corresponding maps.

The two maps are the generators of the modular group, the group of transformations $x \mapsto (ax + b)/(cx + d)$ with whole-number entries and $ad - bc = 1$, which also acts on Stern–Brocot’s tree and on the upper half-plane. The fraction of a tangle is the image of 0 under the product of its moves. What Conway proved is the converse: the fraction determines the tangle, so tangles with the same fraction can always be deformed into each other with their four ends held fixed.

The figure finds the shortest dance from 0 to $17/5$ by breadth-first search over sequences of the two moves: eleven moves, three twists, a rotation, two twists, another rotation and four twists, passing through $3$, $-1/3$, $5/3$, $-3/5$ and on up to $17/5$. Then it untangles. The dancers cannot subtract 1, but a rotation turns a positive fraction negative, and twists then raise it towards 0. The rule “rotate when positive, twist otherwise” brings any fraction back to 0, and the steps it takes are Euclid’s algorithm in disguise: each run of twists subtracts as many 1s as fit, and each rotation takes a reciprocal. When the fraction reaches 0, the ropes are parallel again and come apart.

From fractions to knots #

Join the two top ends of a rational tangle together and the two bottom ends together. The result is a closed curve or a pair of closed curves in space — a knot or a two-component link. For the tangle $p/q$ it is called $K(p/q)$, and it is a knot when $p$ is odd. These are the two-bridge knots: knots that can be drawn so that there are exactly two maxima in some direction, which is what the four strings and their caps in the hero show.

The drawing comes straight from the continued fraction. Writing

$$\frac{17}{5} = 3 + \cfrac{1}{2 + \cfrac{1}{2}} = [3, 2, 2],$$

the knot is three crossings between the middle pair of strings, then two between the upper pair, then two between the middle pair again, with the twists alternating in sense, and caps joining the strings in pairs at both ends. A computation that follows the strings through the crossings and round the caps confirms that the result is a single closed loop. The number of crossings is the sum of the continued fraction’s terms, seven here, and because the diagram is alternating — over and under in turn along every string — Tait’s conjecture proved by Kauffman, Murasugi and Thistlethwaite says that no diagram of the knot has fewer.

The procedure also runs backwards. Given a two-bridge knot drawn as twist blocks, read off the numbers of crossings in the blocks from the end, assemble the continued fraction, and evaluate it: the fraction is the knot’s name. Different drawings of the same knot can give different continued fractions — $[2, 1, 1, 2]$ and $[2, 1, 1, 1, 1]$ are the same number, and reading the blocks from the other end inverts the denominator modulo $p$ — but they always give fractions in the same Schubert class. So a reader with a drawing and a pencil can decide whether two two-bridge knots are the same by computing two fractions and one congruence, a decision that for knots in general needs heavy machinery.

Every knot in the tables with seven or fewer crossings is a two-bridge knot, so every one of them is named by a fraction. The trefoil is $3/1$, a single block of three twists, which makes it also a torus knot. The figure-eight is $5/2$, two twists and two. The cinquefoil $5/1$ and the three-twist knot $7/2$ have five crossings each. The stevedore knot, $6_1$ in the tables, is $9/2$, and $6_3$ is $13/5 = [2, 1, 1, 2]$, four blocks.

Why the fraction is an invariant #

The easy half of Conway’s theorem — that the fraction is unchanged when a tangle is deformed — can be seen through colourings. Colour the four ends of a tangle with numbers, in the manner of Fox, so that at every crossing the over-strand’s number is the average of the two under-strands’ numbers. For a rational tangle, the colours on the four ends are not independent: they satisfy one linear relation, and the ratio in that relation is the fraction. A twist changes the relation by adding the difference across the right-hand ends, which adds 1; a rotation exchanges the roles of the pairs and introduces a minus sign, which takes $-1/x$. Since the three Reidemeister moves preserve colourings, they preserve the relation, and so the fraction.

The hard half is the converse: that two tangles with the same fraction can always be deformed into each other. Conway’s proof, and the later ones by Jay Goldman and Louis Kauffman, show that any rational tangle can be brought by deformations to a standard form read off from the continued fraction, so that two tangles with the same fraction have the same standard form. The standard form is exactly the twist-box picture of the hero.

When two fractions name one knot #

Different fractions can give the same knot. Schubert proved that $K(p/q)$ and $K(p'/q')$ are the same knot, allowing mirror images, exactly when $p = p'$ and $q' \equiv \pm q^{\pm 1} \pmod p$: the denominator may be negated, which takes the mirror image, or inverted modulo $p$, which corresponds to turning the knot over so that the other pair of caps is on top.

The twelve fractions with numerator 13 fall into four groups. $13/2$, $13/6$, $13/7$ and $13/11$ are one knot, because $2 \times 7 = 14 \equiv 1$ and $11 \equiv -2 \pmod{13}$. The four knots have 13, 8, 7 and 6 crossings in their reduced diagrams, and they include $6_3$ as $13/5$.

The numerator $p$ is itself a familiar invariant. It is the determinant of the knot, the number that Fox colourings count: a two-bridge knot $K(p/q)$ can be coloured with residues modulo $p$ in exactly $p^2$ ways in Fox’s sense, $p$ times as many as the trivial colourings. So all four knots in the table have determinant 13, and it is the denominator, up to Schubert’s rule, that tells them apart. The figure-eight $5/2$ and the cinquefoil $5/1$ both have determinant 5; their fractions are in different classes, and so they are different knots, which their Alexander polynomials confirm independently.

Counting the two-bridge knots #

Schubert’s rule makes the two-bridge knots countable by computation. List every fraction $p/q$ with $p$ odd and $q$ prime to $p$, keep one fraction from each class, and record its crossing number as the sum of its continued fraction’s terms.

The counts are 1, 1, 2, 3, 7, 12, 24, 45 and 91 for 3 to 11 crossings, matching the tables and the formula of Claus Ernst and De Witt Sumners from 1987. The numerators needed are bounded, because the continued fraction with the most crossings for a given numerator is all 1s, so a knot with $c$ crossings has numerator at most the Fibonacci number $F_{c+1}$, and the search up to 150 covers 11 crossings. The counts roughly double with each crossing, and the formula shows they grow like $2^{c-3}/3$. All knots grow much faster, so two-bridge knots become a vanishing fraction of the tables: all of them up to seven crossings, 12 of the 21 at eight, and a smaller share at every size after.

Two-bridge links and the even numerators #

When the numerator is even the closure is a link of two components, and the fraction tells how they wind round each other. The simplest, $2/1$, is two circles linked once, the Hopf link; $4/1$ is two circles linked twice, a torus link. In general the linking number of the two components of $K(p/q)$ with $p$ even can be read from the continued fraction too, and the figure’s check that $4/1$ closes into two components is the plat picture’s version of the parity rule: an even numerator leaves the two strings that start at the top on separate loops.

Two-bridge knots are also the knots for which most invariants can be computed in closed form from $p/q$: the Alexander polynomial, the signature, the genus of the smallest surface the knot bounds, and whether the knot is amphichiral — equal to its own mirror image — which by Schubert’s rule happens exactly when $q^2 \equiv -1 \pmod p$. The figure-eight, $5/2$, is amphichiral because $2^2 = 4 \equiv -1 \pmod 5$; the trefoil, $3/1$, is not, because $1 \not\equiv -1 \pmod 3$. A question that is hard for knots in general is a congruence for these.

The dance on the Farey graph #

The Farey tessellation joins two fractions by an arc when $ad - bc = \pm 1$. A twist moves a tangle’s fraction along an arc, from $x$ to $x + 1$, and the rotation is a symmetry of the whole graph. The tangle $17/5$ is reached from 0 by walking along arcs through the convergents of its continued fraction, 3 and $7/2$, each a Farey neighbour of the next. So a rational tangle is a point of the Farey graph, and a sequence of dance moves is a path in it; Conway’s theorem says that the point is all that survives of the path.

This is the surprising connection. The Farey graph was built to describe how well fractions approximate irrational numbers, and the continued fraction is Euclid’s algorithm written down. Neither has anything to do with knots. Yet the most natural family of knots — those made by twisting two strings — is classified by exactly this arithmetic, and the move that untangles them is a step of Euclid’s algorithm.

Tangles in a cell #

The same fractions turned up, unexpectedly, in molecular biology. Circular DNA molecules are knotted and linked by enzymes that cut the double helix, pass one segment through another and reseal it. In the 1980s biologists could see the products under an electron microscope — knots and links of definite types, often two-bridge — but not the mechanism. Claus Ernst and De Witt Sumners, the same pair who counted the two-bridge knots, modelled an enzyme acting on DNA as a tangle operation: the enzyme grips a ball of the molecule, replaces one rational tangle inside it by another, and leaves the rest alone. Equations between tangles, solved with Conway’s fractions and Schubert’s classification, then determined which tangles the enzyme must be exchanging to produce the observed products.

For the recombinase Tn3 resolvase, whose successive rounds of action produce the Hopf link, the figure-eight knot, the Whitehead link and a six-crossing knot in turn, the tangle equations had essentially a unique solution, and it agreed with the enzyme’s structure determined later by crystallography. Knotting is common in long molecules for statistical reasons — almost every long loop is knotted — but the knots an enzyme makes are not random, and their types carry information about the enzyme that the fractions decode.

Drawings matched to tables by Schubert’s theorem #

The plats are drawn from continued fractions and checked to close into one loop; the identification of each drawing with the knot of the same name in the tables is quoted, through Schubert’s classification, not established by comparing diagrams. The census depends on the classification theorem and on the formula for crossing numbers of alternating diagrams, both quoted; what is computed is the grouping of fractions and the sums of their terms. And Conway’s theorem itself, that equal fractions mean equivalent tangles, is not proved here: the dance shows that the fraction changes predictably, which is the easy half.

Fractions counted, Conway’s theorem quoted #

Computed here: continued fractions and Schubert classes of every fraction $p/q$ with $p$ odd up to 150; the counts by crossing number, 1, 1, 2, 3, 7, 12, 24, 45, 91 for 3 to 11, matched against the published values; the plats, with their components traced through the crossings and caps, one for odd numerators and two for even; the shortest dance from 0 to $17/5$ by breadth-first search, and the untangling by the rotate-or-twist rule; and the convergents of $17/5$, checked to be Farey neighbours.

Quoted: Conway’s theorem on rational tangles of 1970; Schubert’s classification of two-bridge knots of 1956; Tait’s conjecture and its proofs; the formula of Ernst and Sumners; the identification of the named knots.

Still open: which knots are close to two-bridge #

The two-bridge knots are completely understood: classified by a fraction, counted by a formula, their invariants computable from the continued fraction. Most knots are not two-bridge, and a natural question is how far a knot is from being one. The bridge number — the least number of maxima in any position of the knot — generalises the two, and computing it for an arbitrary knot is hard: no efficient general method is known even for deciding whether a knot has bridge number three. The tunnel number, the fewest arcs that must be added to make the complement of the knot a handlebody, is one for every two-bridge knot and for many others, and the knots of tunnel number one are classified only partially. Conway’s square dance has generalisations to tangles with more strings, the algebraic tangles, but no fraction-like complete invariant is known for them, and what replaces the Farey graph for three strings is an open question in the theory of mapping class groups.

The fraction is the tangle #

Every tangle made by twists and quarter-turn rotations carries a fraction, the moves act on it as $x + 1$ and $-1/x$, and two such tangles are the same exactly when their fractions are equal. Closing the tangle $p/q$ gives the two-bridge knot $K(p/q)$, drawn from the continued fraction with as many crossings as the sum of its terms; Schubert’s rule $q' \equiv \pm q^{\pm 1} \pmod p$ says when two fractions give one knot, and grouping fractions by it counts the two-bridge knots — 1, 1, 2, 3, 7, 12, 24, 45, 91 from three crossings to eleven. Untangling the ropes is Euclid’s algorithm, walked backwards along the Farey graph.