Knot & Network Lab

The knot's own polynomial — its roots, drawn in the complex plane (the unit circle is amber)

The atlas — one row per p, one square per q · cyan: one single loop · red: comes home early and splits into separate loops (darker = more pieces) · click a square

What you are looking at. The traveller goes p times around the doughnut's hole while turning q times around its tube, then checks whether it is home. If p and q share no factor it is home only at the very end, and the path is one single loop — a torus knot, written T(p,q). If they share a factor d, it comes home after only p/d laps and q/d turns, and the full pattern falls apart into d separate loops, each a smaller knot; the faint loops are the parts it never visits. So the rule is the same as the prime loops on the sphere — and a prime p makes a single knot with every q that is not a multiple of it. What the torus adds is knotting: when both numbers are at least 2 the loop cannot be untangled without cutting it. T(2,3) is the trefoil, T(2,5) the cinquefoil, and the fewest crossings any drawing of T(p,q) can have is the smaller of p(q − 1) and q(p − 1). T(p,q) and T(q,p) are the same knot. Drag to turn, scroll to zoom, double-click to reset.

Lossless or leaking — the pitches in the complex plane · real part: the pitch · below the line: how fast it dies away

Echoes — ring every pitch together and listen at each length x · ticks: log n (cyan: prime · violet: prime power · amber: composite)

The same test on the zeros of ζ — ring the first 2,000 zeros together

The cable. Choose "separate strands" and the loops are no longer tied together: each one rings on its own, like the wires in a cable. A closed path then has to stay on a single strand, so the echoes fall only at one loop's length and its multiples. With strands of length log p that means the primes and their powers, with the composites silent, the same pattern the zeros of ζ give below. Physicists call this bundle the primon gas: the strands are the primes, every way of exciting several strands at once corresponds to exactly one whole number, and adding up all its states gives ζ as the product over the primes. Its limit is the product's limit: it only settles on shells bigger than 1, and each strand's own pitches are not the zeros of ζ.

The network. Each loop is a string whose two ends are tied to one shared junction; a wave reaching the junction splits into every loop and comes back. Its pitches solve an exact equation — the sum over loops of tan(k·L/2) = 0 — plus each loop's own notes 2πm/L, which have a still point at the junction. The echoes. Ringing all the pitches together and listening at a length x is a Fourier transform; it peaks at the lengths of the closed walks through the junction (the network's trace formula: Roth 1983; Kottos and Smilansky 1997). With loops of length log p, a walk around loops 2 and 3 has length log 2 + log 3 = log 6 — so the network echoes at composite numbers too: nothing cancels. The zeros. Ringing the zeros of ζ the same way gives Riemann's explicit formula: deep echoes at log p for every prime, softer ones at prime powers, and silence at composite numbers — they cancel. That cancellation is exactly what makes ζ special: its "network", if it exists, would have to be one in which only the primes and their powers ever echo. The readouts measure the echo strength at each kind of number against the background.

The zeros in the s-plane — the graph's own critical strip, turned on its side · across: the height t · up: the real part σ · amber: the critical line σ = ½

All closed walks of each length — split into the prime cycles and the repeats of shorter ones (log scale) · amber line: qⁿ, which the prime cycle theorem says n × (prime cycles of length n) approaches

Prime cycles. On a network, a closed walk that never turns straight back on itself is a cycle; one that is not just a shorter cycle traced several times is a prime cycle — your definition, and the standard one. Count every closed walk of length n (grey), and some of them are repeats of shorter prime cycles (red). Removing the repeats with the Möbius signs — add the walks of length n, subtract those of length n/p for each prime p dividing n, add back those of n/(pq), and so on — cancels every repeat and leaves exactly the prime cycles (cyan). This is the same cancellation that isolates the primes among the whole numbers. The prime cycle theorem. On a network where every point has q + 1 links, the number of prime cycles of length n grows like qⁿ/n — the prime number theorem's shape, with qⁿ in place of x. This is the discrete version of the saddle: every junction branches, so paths multiply exponentially, just as closed straight paths do on a saddle-shaped surface. A Riemann Hypothesis that is true. Every network has a zeta function built as a product over its prime cycles (the Ihara zeta function). Its "zeros" are the numbers plotted on the right. The network's Riemann Hypothesis says all of them, apart from a few trivial ones, lie exactly on the circle of radius √q — and it holds exactly for the networks called Ramanujan graphs (Ihara 1966; Sunada; Lubotzky, Phillips and Sarnak 1988). The Petersen graph, K₄, K₅ and the cube all pass. The eigenvalues come from the network's own adjacency matrix through the Ihara–Bass formula.

The same torus cut open and laid flat (left) · every prime's wave as a dial (right)

What each traveller reads as the height climbs, on the shell you chose · 0° is a touch at the south pole (a zero)

The pull meter · add the primes one at a time and watch where the value stands, at the height and shell you chose

Tune the zeros · the torus stays as it is · stretch every touch height together and see whether the primes still ring

What fits · how much of the primes' ringing survives, for every stretch from 0.8 to 1.25

Tuning the zeros. The first 2,000 touch heights of ζ are rung together as pure tones and listened to along the logarithm of the numbers, where each prime's loop is a plain length, log p. With the true heights the echoes land on the primes and their powers and nowhere else. The slider multiplies every height by the same factor. That slides every echo to a new place: the echo of p moves from log p to log p ÷ stretch, so it no longer sits on a whole number. The lower chart measures, for every stretch, how much ringing is left at the true primes. It is 100% at a stretch of exactly 1 and close to nothing elsewhere. The small bumps away from 1 are single coincidences, where one number's echo happens to slide onto another's place; no stretch other than 1 lines up more than one at a time. Scattering the touches at random blurs the echoes, the largest primes first. Why the logarithm. Taking the logarithm undoes the multiplying: numbers that multiply become lengths that add, and in that stretched-out scale the zeros are steady tones. This panel is drawn in that scale.

The pull meter. Each prime adds its push or pull to a running tally, in the order 2, 3, 5, 7 and so on. Each line shows where a traveller's value stands after that many primes, up to the first 5,000, at the current height. The dashed line is the true value, where there is one. On a shell bigger than 1 the tally settles after a handful of primes. Between ½ and 1 it settles more slowly. On the ½ shell there are two behaviours. At an ordinary height the tally hovers around the true value without ever quite resting. At a touch it keeps sinking toward 0° for as long as primes are added, but so slowly that 5,000 primes only bring it to within a few degrees. No fixed number of primes reaches the south pole; it takes all of them. The look-alike has no tally of its own, because it is a blend of the twins' values and not a traveller: at its touch on shell 0.8085 both twins' tallies level off about 70° away.

What you are looking at. Every prime's wave is somewhere in its cycle, and "somewhere in a cycle" is a point on a circle. Two primes give two circles, which together make a doughnut: one prime's angle runs around the hole, the other's around the tube. As the height climbs, both angles advance at once, at the rates log p and log q, so the height traces a path that winds around the doughnut. In the torus-knot mode a path comes home when its two winding numbers are whole numbers. Here it comes home only if some power of the first number equals a power of the second. Try 4 and 8: 4³ = 8², and the path closes into a trefoil. For two different primes that can never happen, because a number factors in only one way. So the path never comes home, and it slowly covers the whole doughnut. With every prime included the torus has one circle per prime; the dials show the first twelve of them, and the doughnut shows any two.

In step, and the other starts. Every traveller moves at the same speeds, so their paths are parallel and never cross. They differ only in where they start. ζ's traveller starts with every prime at 0° together: all in step at height 0. A typical start puts every prime at its own random angle. The twins start from angles set by each prime's remainder on division by 5 (remainder 1: 0° · remainder 2: 270° for twin A, 90° for twin B · remainder 3: the other way round · remainder 4: 180°; the prime 5 has no circle for them). The twins are mirror images of each other, and each is a known function with its own mirror rule. A composite number never adds a new circle: its angle is the sum of its primes' angles (try 2 and 6).

Reading the value. Each point of the torus gives a value: multiply together one factor per prime, where a prime pushes the value away from zero when its dial points right and pulls it toward zero when its dial points left, with strength 1 ÷ p^σ. The flat map is shaded by that push and pull for the two chosen numbers alone. The graph shows what each traveller reads along its path, as a distance from the south pole. Solid amber is the true ζ and solid red is the true look-alike, both computed exactly. The violet curve and the dashed amber curve are products of the first N primes, so they are approximations: on shells bigger than 1 they settle, between ½ and 1 they are close but never finished, and at ½ they do not settle at all.

What is known. For almost every starting point on the torus, the traveller never touches the south pole on any shell bigger than ½ (Helson, 1969). That cannot vouch for one particular start, and ζ's is one particular start. The look-alike is not a traveller at all: it is a fixed blend of the two twins, so its value is not read at any single point of the torus. On shell 0.8085 at height 85.70 the twins are the same distance from the south pole, about 70°, and point in directions that cancel in the blend; the blend touches while neither twin does. The twins and ζ each have both the torus and a mirror rule. The look-alike keeps the mirror rule and loses the torus. A typical start keeps the torus and has no known mirror rule.