Hildemar's Knots and the Einstein's Revenge Cluster

Intelligent species living inside a neutron star in the Einstein's Revenge cluster

Knotbot 'Loopy'
Image from Liam Jones
A virch model of 'Loopy', the first physiologically accurate drone used to interact with the Knots. Note that Knots, lacking a visual sense, would never see each other this way..


The Einstein's Revenge cluster is a group of 6 rapidly rotating neutron stars spread across 11 lightyears in Sagittarius, 6201 lightyears from Sol.

The superfluid outer core of every star in the cluster is home to an ecosystem based around quantum turbulence, with lifelike processes arising from complex knots of vortex filaments. One star, KJEI-54-458945, is home to a xenosophont species, Hildemar's Knots.


The Einstein's Revenge cluster was first reached in the late 9800s by the Hrupta Investigatory Fleet, STC. The 6 neutron stars of the cluster had been observed to display unusual glitch behaviour a millennium previously. The Investigatory Fleet consisted of six solid state ships, each with a crew of three thousand uploaded Sapient Hyenas (the dominant clade of Hrupta) led by a S:1 matriarch.

Upon arrival, the ships set up Lightway links between themselves before turning their attention to the neutron stars.

The Investigatory Fleet used the Geminga Bootstrap Method to investigate the stars' interiors. In the first stage, each ship used high energy gamma ray lasers to inscribe structures on its star's crust. Transceivers holding AIs kept in contact with the ship, while mobile probes made of patterns of crystal defects could travel deeper into the crust and report back with their findings. The crystal defect probes built a second set of structures in the nuclear pasta region, which in turn built structures from vortex filaments which would explore the superfluid core.

The first two stages revealed evidence of advanced technology. Both the crust and the nuclear pasta layer hold remnants of complex structures made from topological defects and braided nucleii — these are clearly artificial, but the remnants are too degraded to offer any evidence as to their original purpose.

Final, explorations into the core revealed the existence of a biosphere composed of vortex filaments in every star. As the Investigatory Fleet shared their findings, it became clear that the biospheres share a common ancestry, with evolutionary dating suggesting a point of divergence 1.3 million years ago.

Hildemar's Knots, a sophont species, were discovered in 9837. The fleet studied the species first by remote observation by sensors in the pasta layer and drones imitating local wildlife. In 9840, the fleet used drones imitating the Knots themselves to interact with and study them in depth.

As of the present day, the Revengian Investigatory Authority has yet to establish true contact, despite numerous petitions from the Eden Institute. However, an ecosystem template, including plans for the Knots themselves, was leaked in 10350. The STC is believed to have an agreement with the Caretaker Gods regarding protection of the Revengian biospheres, but the details remain obscure.

Plans are underway to provolve a viable candidate (Hybrachian Braids, a species of subsapient social browsers) on a separate Revengian star.

The Revengian Ecosystem


In the superconducting superfluid neutron mantle, the star's rotation is quantised, forming a dense array of neutron vortex filaments, each of which carry a single unit of angular momentum. The magnetic field is similarly quantised into proton vortex filaments, each carrying a unit of magnetic flux. Attractive and repulsive forces, depending on the circumstances, govern how the vortex filaments interact.

In simple models, neutron and proton vortex filaments arrange themselves in a regular array of straight lines running from one side of the star to the other, aligned with the axis of rotation and magnetic field respectively. However, with the onset of quantum turbulence, they can become tangled in immensely complex ways.

Neutron stars have an excess of left-handed electrons compared to right-handed electrons, produced though the Urca process during their formation. These electrons form a spin current travelling along proton vortex filaments. If the filament forms a loop, the left-handed electrons can be trapped and stored. Spin currents can change the force between filaments. If two filaments of the same type touch, they will reconnect, changing their topology. This releases energy in the form of Kelvin waves, which travel down the filaments.

Revengian life arises from the interaction of all these elements. Information is stored in the topology of complex braided filaments. Simpler braids all store energy, and transfer it through reconnection events. Left-handed electrons act a something like an essential mineral, helping govern filament interactions and reconnections. Kelvin waves provide fast signalling, rather like action potentials in Earth life..

The Revengian analogue of the cell is the thread, a group of vortex filaments capable of self-replication. A thread might be less than 10 nanometres across but can reach almost a metre long. Most form loops, though a few a linear.. Single-thread organisms fill many of the same ecological roles as bacteria and protists. More complex organisms, including Hildemar's Knots themselves, are composed of bundles and braids of specialised threads.

Interactions between filaments act at a different rate than chemical interactions. Revengian biological processes have length and time scales roughly 50 times smaller and faster than the Terragen equivalent.

Dead Turbulence

Most vortex filaments in the Revengian environment are not part of living systems. While the neutron superfluid itself it completely frictionless, this "dead turbulence" forms a sort of viscous medium for Revengian organisms. As they move through it, they will intersect vortex filaments, draining their energy.

Dead turbulence also provides a medium, albeit a highly dispersive one, for the transmission of Kelvin waves. The turbulence is non-isotropic. On average, more filaments are oriented along the star's magnetic field and axis of rotation. Organisms and Kelvin waves will move more easily along the direction of the filaments.


The Revengian ecosystem is powered by the differential rotation between the interior and crust.

Neutron stars spin down gradually due to magnetic dipole braking. As this happens, the crust slows more rapidly than the interior, and the two become decoupled. A glitch occurs when neutron vortex filaments become unpinned from the nucleii, expanding outward in an "avalanche", annihilating as they hit the crust, and thereby transferring the angular momentum of the interior to the crust. An outside observer sees the star's angular velocity increase suddenly.

Producers, or rotatrophs, gather energy by unpinning neutron vortex filaments from the crust in a more controlled manner. As the released vortex filament moves outward, it intersects special organs in the rotatroph, causing reconnections and transferring energy. Most rotatrophs, therefore, need be anchored directly to the crust, and structure resembling ladders aligned parallel to the axis of rotation. Parasitic rotatrophs extract energy from filaments unpinned by others. Rotatrophs gather in "ladder forests" around the equator.

Consumers can gain energy in various ways. Some directly absorb the braids of other organisms. Some break apart these braids and absorb energy through reconnection. And a small class of parasites work by leeching energy from spin currents and Kelvin waves.

The Chromatic Biosphere

In the inner core of neutron stars, the neutron superfluid changes phase to become an exotic colour-charge superconductor. This environment holds a separate biosphere, based on colour vortex filaments, extracting energy from the temperature gradient, and mediating biological processes through hyperon synthesis. Organisms are relatively simple, corresponding to prokaryotic analogues. It is still unknown what relation these chromatic biospheres have to other Revengian life.

Knotbot extending an arm
Image from Liam Jones
Virch image of 'Loopy', an artificial construct designed by Terragens to interface with Hildemar's Knots. Here the knotbot is extending an arm, consisting of so-called 'nuclear pasta'. This robot resembles a typical Hildemar's Knot individual very closely.

Hildemar's Knots


Evolution; It remains a matter of debate whether the Knots evolved naturally. However, a highly plausible evolutionary history is consistent with genomic analysis and multiple S2 simulations.

The dominant phylum on the Knots' star contains animals with many thousands of long, thin tendrils, which contain most of the nervous system and provide a highly effective means of perceiving and interacting with the world around them, and communicating with others.

The Knots appear to have evolved from a class of communal filter feeders. These animals move side-by-side in large groups, extruding a fine mesh of tendrils that link them together and allow them to communicate. The mesh constantly twists and reforms to find the easiest path through the dead turbulence while ensnaring the Knots' food source — a plankton analogue consisting of organisms and detritus tens of micrometres across. The feeding process requires complex co-ordination betweens hundreds of Knots.

When found, the plankton is superabundant, providing more than enough energy for the entire group. However, different types thrive only in specific environmental conditions. A combination of identifying these conditions and complex social behaviours drove the Knots towards intelligence. Finally, the ability to manipulate their environment, by culling or promoting various species, and by arranging dead turbulence, gave them to final push towards sophonce.

Anatomy: The main body is looped into a simple torus knot, either trefoil, cinquefoil or septafoil. The crossing number is developmental rather than genetic — offspring share their mother's linking number. All Knots are right-handed.

The body is 2mm thick and forms a knot 2 to 4 centimetres across when fully extended (for the same body volume, trefoils are about twice the diameter of septafoils). Though constrained by their topology, Knots are very flexible and can elongate, contract, or assume complicated shapes.

A bundle of 60-90 flexible limbs are coiled round the main body. To reach out, a Knot unwraps a limb to the desired length. Each limb is itself a coiled bundle of digit fibres, and can be divided further into sub-limbs of various sizes. Finally, each digit is ciliated by individual thread-cells. A Knot may have as many as 10,000 digits, each with 10,000 cilia. It is common for digits to be damaged or severed, and they can be quickly regrown. Fully extended, a limb can reach 42cm in all directions.

The limbs are essential to feeding. When extended, they form the mesh that filters plankton. When feeding, cilia will ensnare individual plankton and carry them back to the sub-limb, which "digests" the plankton by trapping it in a microscopic braid of threads and extracting its rotational and magnetic energy.

Locomotion Dead turbulence is a viscous medium through which Knots can swim — but the similarities with water end there. Turbulence is not isotropic. To move through it effectively, sliding through it where the filaments are parallel to the direction of motion and friction is lessened, and pushing against it where the filaments are transverse and friction is at its greatest. The resulting motion is highly complex, a difficult motor task requiring dynamic adjustment at the micrometer scale, and has as much in common with climbing or burrowing as swimming.

Senses: The Knots' primary sense is tactile. With their cilia and digits extended they can form an extremely detailed picture of their local environment, including the state of the local turbulence. Most Knots will actively investigate their environment with their digits even when resting.

Long-distance sensing is more complicated. No single sensory medium is equivalent to vision — Knots are effectively blind — but they use a variety of strategies. Kelvin waves can travel for long distance, but are highly dispersive, travelling only along filaments, and are insufficient for forming images. Sound and second sound (composed of fast waves of heat) travel more effectively, but only weakly interact with the Knots biology. Finally, a Knot can use the state of the local dead turbulence to make inferences (different animals leave different traces in the turbulence).

These sensory modalities are faintly equivalent to human auditory and olfactory senses. They are more limited than touch, but essential in providing Knots with information beyond their tactile horizon.

Lifespan and Reproduction: Knots are protandrous sequential hermaphrodites. All individuals are born male and remain so until sexual maturity, 73 days later. Thereafter, they can change sex depending on complex social and dominance cues. In general, the more dominant and socially adept individuals become female, but it is also possible for a more dominant Knot to compel a lower-ranking ally to become female. Coalitions of lower ranking Knots may also compel more dominant individuals to revert back to being males. Monogamous, polyandrous and polygynous mating patterns are all present. Sexual politics among Knots is understandably complicated.

Mating occurs during interleaving. The fertilised bundle of threads grow from a a digit root and trace the length of the mother's body until they reach their starting point, where they join the same root. Then they disentangle themselves by splitting and rejoining at each crossing in turn, until the only point of attachment is the original digit root.

Two things follow from this process. First, it ensures that the offspring almost always has the same linking number as its mother. However, any errors in the disentangling process will result in the offspring having a different linking number, or — far more often — the death of the offspring. Second, the offspring will directly inherit some of its mothers digits, and hence neural tissue.

After disentangling, maturation takes ten days. During this time, it will interleave regularly with its mother and intermittently with other members of the group, and begin to feed itself. Once mature, it separates away from the root.

Knots have an average lifespan of two years.


Rather than action potentials, Revengian life uses Kelvin waves as nerve impulses. The slower spin currents, meanwhile, act in a role analogous to endocrine signalling.

Knots have a highly distributed nervous system. The majority of neural tissue is the limbs, with the central body providing little more than routing and basic executive control. The limbs are largely autonomous, providing everything from co-ordinated motor activity and sensory integration to basic problem-solving.

They are liable to act independently. One set of limbs may react a situation aggressively, another calmly, and another submissively. Knots devote considerable time to training their limbs to act appropriately (although it would be more accurate to say the limbs train one another) and to using different sets of limbs in different circumstances. In extreme cases, this may result in a Knot forcefully tearing out its own limb so a more amenable one can grow. More generally, a Knot can keep various elements of personality "in storage" by not using the limbs where these traits are held.

Limbs can even communicate among themselves without going through the central body. Any direct contact between the cilia can be used to relay Kelvin wave signals. This communication is essential in keeping behaviour coherent when the limbs are fully extended.

A contracted Knot, with most of its limbs coiled around the main body and communicating with one another, is deep in thought. By shifting and rearranging the connections, it can gain new insights into its own thoughts and produce bursts of creativity.

Limbs of different individuals can also communicate with one another. When a group of Knots are linked in a feeding mesh, the linked limb tips of two individuals may actually be able to act as a more coherent entity than either individual.

Interleaving is the core of all Knot social interaction. Two or more individuals contract with their limbs entangled together and sharing nerve impulses in hundreds of places at once. The resulting communication is nonlinear. Thoughts, memories, and even personality traits diffuse between the two individuals. Interleaving even allows one Knot to train the arms of another. While intimate, interleaving is not always a positive experience. Sometimes it is part of agonistic interaction. A Knot may compel another to interleave, and even use the process to switch the sex of and/or partially rewrite its opponent. Knots are perfectly capable of lying through interleaving, despite its apparent intimacy. Interleaving and more distant arm contact is the basis of language. Individual elements consist of actual motor signals (which may be indicative or imperative), sensory impressions, and deeper thoughts, linked by a deep structure of logical and modal connections.

The Knots' ecological niche as co-operative filter-feeders affects their psychology. Because they lack good distance senses, remaining in direct physical contact with others while in a feeding party is essential.

They are extremely good at quickly co-ordinating complex behaviours on the fly. The extends beyond feeding. They are socially adept, and a Knot introduced into a new group will quickly pick up the behaviours and even personal traits of that group (and introducing some of its own if it is sufficiently dominant). They are not particularly individualistic, and have no problem with adapting their personality to the standards of the group. They are not territorial, but can become tribal. Still, even groups who have just met one another are adept at finding a common goal and working towards it. Nevertheless, this does not prevent interpersonal struggles — Knots still compete for mating rights and to give their input into social matters more weight. It's just that such struggles take place within a communal, closely-knit social context.

Culture and Society

Knots have a complex and obscure society based around the circulation of self, orthodoxy and noise.

During interleaving, Knots can gain personality traits from one another. One might obtain introversion, a heightened aggression, or different sexual preferences. In current Knot culture, this practice has become highly formalised. Certain constellations of traits are organised into "formulaic elements of self" which can be passed from one Knot to another, thus giving rise to a circulation of self.

Young Knots are trained extensively and almost forcefully to pass on elements of self without disturbing them. In most cases, the individual has no freedom to modify elements, only to assemble them in particular ways. (It is possible to modify the elements, merely taboo.) The dynamics of circulation are very complex. Individuals may use self as a currency for mating, travel or political access, or they may give self according to various cultural dictates. One may try to acquire particular elements to accomplish a particular task, for prestige, or merely to try experimenting with new combinations.

Circulation of self is generally divided into two facets. On one side is orthodoxy. Groups of Knots in close contact, especially if they feed together, tend to quickly converge upon a particular way of acting and thinking which may seem incomprehensible to outsiders. This orthodoxy is generated from the initial personalities of the various members, and can itself be circulated to others.

The other side is noise. Through the ritual of "submitting to chaos", a Knot will contract and rearrange its limbs randomly — perhaps with the help of associates helping to shuffle the limbs, or by wandering through the forest taking environmental cues. Diverse traits and tendencies of various limbs are brought into contact in novel ways. At the end of the ritual, the Knot emerges with an entirely new personality, with entirely new elements of personality.

The dichotomy of orthodoxy and noise is universal in Knot society and seems to be considered fundamental by the Knots themselves. Orthodoxy is the public face of the individual, essential to taking part in and contributing to society. It is appropriate and represents order. Noise is the private face of the individual, essential to generating genuine novelty. It is inappropriate, taboo, even shameful, and represents discordance. Rituals surrounding noise and submitting to chaos usually involve a theatrical breakdown of the social order, in which newly noise-laden individuals are allowed and even encouraged to act against the social order.

Feeding groups, consisting of twelve to sixty well-synchronised Knots, are the basic elements of community. Social organisation is fluid — individuals may leave and join groups as their personality shifts. Some groups may share members as part of the circulation in self. Two groups in close contact merge easily, and groups may split without rancour as they become two large to sustain a coherent orthodoxy. Some small groups may even be used as elements of trade for larger groups.


All Knot cultures use dead turbulence as an artistic medium. The artist sculpts the turbulence on a microscopic scale to form aesthetically pleasing patterns. A Knot can view the artwork either by lightly touching it with cilia, or by swimming/crawling through it. Repeated viewings tend to damage the patterns. Some are given constant maintenance by the artists; others are allowed to dissipate.

Turbulence art may be combined with a sort of gastronomy, where specialised types of plankton with different "tastes" are released into the sculpture. The structure of the turbulence governs how the plankton move and where they are encountered.

Personality art is common. Knots can acquire formulaic elements of self and arrange in odd and interesting ways. The resultant combinations are usually unsuitable in day-to-day living and are contained in unused limbs that can be brought out for a display.

In other cases, entire communities may become an artwork. To experience the full effect, a Knot must spend time with community and become shaped by its orthodoxy. Problems can arise if individuals join and become so closely aligned with the group that they refuse to leave.

Science and technology

Superficially, Knot technology appears to be primitive. Most groups live in the open, with little shelter beyond the ladder forests. But this first impression is misleading. Knots evolved the final step to sophonce by manipulating ecological variables to increase the amount and quality of plankton they could find. This, rather than the creation of tools, has become the technological basis of their civilisation. They do, in fact, have an inituitive understanding of very complex ecological dynamics.

By cultivating some species, culling others, manipulating the quality of the dead turbulence, and shifting nutrients around, they can stifle dangerous predators and competitors and obtain greater volumes of food. The Knots' manipulations are highly sophisticated, balancing tens of thousands of ecological variables across the entire mantle with multiple failsafes and layers of redundancy. Many species have become inadvertently specialised to their roles in the system.

The Knots' mental modifications are intertwined with this structure. By sharing elements of self, they can pass on not only skills and knowledge but also personality traits — most lines of work can be rewarding when one is equipped with the right sort of enthusiasm.

Knots have a advanced ecological science and applied mathematics, with detailed and highly accurate models of their turbulence superfluid environment. They understand that the star is spinning, and that this provides the energy source of their ecology and civilisation. While they have no written language, their ability to quickly absorb large amounts of information means their "oral" knowledge base is more than sufficient.

The current level of technology appears to be static, but there is some evidence that the Knots may have had significantly more advanced technology in the past.


The shared ancestry of Revengian biospheres and the technological remnants present in each star suggests a common link, but the nature of that link remains unknown. Various theories have been put forward, but so far remain unconfirmed.

Some theories claim that the initial Revengian biosphere evolved naturally on one star and was later transplanted to others by a spacefaring xeno civilisation. Other theories go beyond this, suggesting that identical copies of a biosphere were created artificially by a xeno transapient.

Finally, some suggest that the Knots — or earlier Revengian sophonts — colonised the cluster by themselves. By using the Geminga Bootstrap Process in reverse, they may have been able to upload a biosphere template onto a ship and send it to nearby stars. The most outré version of this theory notes that a properly engineered neutron star may provide sufficient energy and toposophic substrate to create a Weylforge in its core, by means of which the Precursor Knots could have linked the cores of the stars with wormholes.
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Development Notes
Text by Liam Jones
based on material by Anders Sandberg
Initially published on 04 November 2001.

Additional Information
Simulations of superfluid vortex filaments. This is the basis of the Hildermar's Knots' biology. If you can imagine something a million times more complex and structured into something orderly, that might give you a sense of what's going on inside their thread-cells.

Towards High-quality Visualization of Superfluid Vortices YouTube: Xiaopei Liu

Superfluid Kelvin Waves YouTube: Lathrop Lab