Ahoy there, sailor!
You’ve been away on a long voyage across the four-dimensional sea. Its heaving surface isn’t a sheet of water but a whole volume of it, slamming into your hull from directions no 3D sailor has a name for. The horizon all around you is a sphere rather than a circle. As well as fore and aft, port and starboard, and up and down, there’s a fourth axis, and your boat can steer along it. Charles Hinton, who spent the 1880s trying to teach people to picture four dimensions, named its two directions ana and kata, and they’ve served 4D sailors ever since.
Now the harbour wall is coming up off your ana bow, and it’s time to moor.
1. The bowline slips out ana
You throw a line around the bollard, tie a bowline and step ashore. Behind you, the knot slides open: one strand steps ana, slips past another, and the bowline falls apart.
A knot holds in three dimensions because rope can’t pass through rope. Draw a knot flat and at every crossing one strand goes over the other. To swap which strand is on top, the upper strand would have to drop past the lower one right where they cross, and there the two would meet. Solid rope can’t do that.
In four dimensions there’s a way around. Give every point of the sea four coordinates, \((x, y, z, w)\). The first three are the usual ones, and \(w\) measures how far a point is ana (\(w > 0\)) or kata (\(w < 0\)). Mathematicians call this space \(\mathbb{R}^4\).
Lift a short piece of the upper strand ana. It’s now at a different \(w\) from the lower strand, so the two can share the same \((x, y, z)\) without touching. Lower it past the lower strand, bring it back kata, and the crossing has changed without the strands ever meeting.
In the figures, colour is \(w\): blue is kata, cream is \(w = 0\) and orange is ana. Two pieces of rope touch only if they’re in the same place and the same colour.
That’s enough to untie any knot. Pick a starting point on a knot diagram, away from any crossing, and walk once around the diagram. At every crossing you reach for the first time, make your strand the upper one. When you reach a crossing for the second time you’re on the lower strand, so leave it as it is.
The new diagram is always the unknot. To see why, give the rope heights that match it. Start at height 1 and let the height fall steadily as you walk, down to 0 as you arrive back at the start, then climb straight up to height 1 there. At every crossing the strand you reached first is higher, just as the diagram says. For each height \(h\) between 0 and 1, the rope now has exactly two points at that height, one on the way down and one on the climb. Join them with a horizontal segment. Segments at different heights lie in different horizontal planes, so none of them meet, and together they fill in a disc whose edge is the rope. Shrinking the rope across that disc turns it into a small circle, so it was the unknot all along.
So every knot turns into the unknot after some crossing changes, and in \(\mathbb{R}^4\) every crossing change is free.
That covers rope lying in ordinary space. A rope anywhere in \(\mathbb{R}^4\) can be moved there first, by shrinking every point’s \(w\) to 0. Two points of the rope would only collide on the way if they had the same \(x\), \(y\) and \(z\), and a slight nudge to the rope beforehand removes any such pairs. So in \(\mathbb{R}^4\), every knot comes undone.
The general idea is codimension, the dimension of the space minus the dimension of the object in it. A rope is 1D. In ordinary space it has codimension 2, which is enough room to go around another strand but not enough to get past it. In \(\mathbb{R}^4\) it has codimension 3, and the spare dimension lets every crossing undo itself.
2. Something to tie to
No knots, then. But a rope doesn’t need a knot to hold. Pass it around a bollard and fuse the ends into a closed loop, and there’s nothing left to untie. You try it on the nearest bollard, and the loop slips off ana. The harbourmaster has watched plenty of visiting sailors do this, and points you down the quay to a different kind of bollard.
To see why the first one failed, start with an ordinary post in three dimensions, and for now assume it goes up forever. A real post has a top you could lift the loop over, and section 4 comes back to that.
Whether the post holds the loop comes down to a count. Stretch a surface across the loop, such as a soap film, and give it a front and a back. Follow the post upward, adding 1 each time it passes through the film from back to front and subtracting 1 each time it passes from front to back. The total is the linking number of the loop and the post. A loop dropped over the post has linking number \(\pm 1\). A loop lying on the quay beside the post has linking number 0.
Two facts make this count useful. First, it doesn’t depend on which film you choose. Two films with the same edge together form a closed surface, and a line running off to infinity at both ends leaves a closed surface as often as it enters. Second, it can’t change while the rope moves without touching the post. Carry the film along with the rope. Crossings in the film’s interior appear and disappear only in pairs of opposite sign, as the post slides over a fold in the film, so they leave the total unchanged. A single crossing can only escape across the film’s edge, and the edge is the rope itself. So no motion takes the loop from around the post to the quay beside it, and the post holds the rope.
The count needs the film and the post to cross at isolated points. In \(\mathbb{R}^n\), an \(a\)-dimensional object and a \(b\)-dimensional one in general position cross at isolated points when \(a + b = n\), and miss each other entirely when \(a + b < n\). In 3D the film is 2D and the post is 1D, and \(2 + 1 = 3\).
In 4D the film is still 2D and a line is still 1D, but now \(2 + 1 < 4\). Nudge the post ana and it misses the film altogether. The rope can then shrink across the film to a tiny loop without touching the post, and float away. To hold the rope, the bollard has to cross the film at isolated points, so it has to be 2D: \(2 + 2 = 4\). In \(\mathbb{R}^4\), a rope loop can only be held by a two-dimensional bollard.
The same count works in any dimension. A \(p\)-sphere is the \(p\)-dimensional version of a circle. A loop of rope is a 1-sphere, and a closed sheet with no holes, like a balloon, is a 2-sphere. A \(p\)-sphere bounds a \((p + 1)\)-dimensional film, so it can be linked with a \(q\)-dimensional object in \(\mathbb{R}^n\) when \((p + 1) + q = n\), or
\[p + q = n - 1.\]A loop of string on a table links a point (\(1 + 0 = 2 - 1\)), a rope in ordinary space links a line (\(1 + 1 = 3 - 1\)), and a rope in \(\mathbb{R}^4\) links a plane (\(1 + 2 = 4 - 1\)).
Real bollards are solid, though. What counts is a bollard’s core, what’s left if you shrink it without it ever touching the rope. An ordinary post shrinks to the line up its middle. The rope never touches the post as it shrinks, so the linking number doesn’t change, and the line holds the rope just as the post did. So \(q\) is the dimension of the core.
The bollard you tried was the obvious 4D version of a post. A 4D sea has three horizontal directions, \(x\), \(y\) and \(w\), and that bollard was round in all three and ran up in \(z\). Its exact shape doesn’t matter, though. Any bollard that’s only so wide in each of \(x\), \(y\) and \(w\) shrinks to a line, and a line can’t hold a rope in \(\mathbb{R}^4\). The harbourmaster’s bollard is different. It’s round only in \(x\) and \(y\), and it runs up in \(z\) and keeps going ana and kata in \(w\), forever in both, so its core is the \(zw\)-plane.
The easiest way to see the difference is to look at the harbour one slice at a time. The slice at a fixed \(w\) is an ordinary 3D space, and moving the slice ana and kata shows how things change along the fourth axis.
The round bollard gets thinner as you move ana, the same way the slices of a ball shrink towards its edge, and then it stops. Beyond its edge the slices are just water. So the rope steps ana past the bollard’s edge, slides sideways, and comes back kata beside it. The long bollard is in every slice, so wherever the rope goes, the bollard is still inside it.
At the long bollard, the harbourmaster brings a coil of rope from the ropewalk, runs it around the bollard and around a cleat of the same long shape on your deck, and splices the ends together where they lie. The splice is necessary. The count that stops a closed loop coming off also stops one going on, so the only way to get a closed loop around the bollard is to close it in place. Your boat is moored without a single knot.
3. Missing knots? Take a tarpaulin
The boat is safe, but every knot you know is now useless. You can’t lash a crate, hitch a fender or tie off a sail. The harbourmaster sees you turning a length of rope over in your hands and passes you a tarpaulin. “If it’s knots you want, give up on rope.”
Go back to the codimension count from section 1. A rope has too much room in \(\mathbb{R}^4\). A sheet has less. A tarpaulin is 2D, so in \(\mathbb{R}^4\) its codimension is 2, the same as rope in ordinary space. Closed sheets in \(\mathbb{R}^4\) can be knotted. From here on a closed sheet means a 2-sphere, as in section 2.
The move that undid the bowline doesn’t work on a sheet. Lifting one patch of sheet past another needs a spare dimension, and a sheet in \(\mathbb{R}^4\) has none. That doesn’t prove any sheet is knotted, since some other motion might still undo it. A proof needs an invariant, a quantity that stays the same however the sheet moves without passing through itself, and that differs between the sheet in question and a plain sphere.
The first knotted surface was built by Emil Artin in 1925, by spinning. It takes three steps to see how.
Spinning in 3D. Hold a semicircle with both ends on a vertical axis and spin it about the axis. Each point of the arc travels around a circle, bigger the further it is from the axis, and the two ends stay where they are. The arc sweeps out a sphere.
Turning in 4D. In 3D, a rotation turns about a line: points on the axis stay put and every other point moves in a circle. In 4D, a rotation turns about a whole plane. A rotation that mixes \(x\) and \(w\) leaves every point with \(x = w = 0\) where it is, which is the \(yz\)-plane, and moves every other point in a circle. A point that starts in ordinary space at distance \(d\) from the plane swings out ana, is at \(x = 0\) and \(w = d\) after a quarter turn, and arrives back in ordinary space on the far side of the plane after half a turn. The second half of the turn brings it back through kata.
Spinning a knotted arc. Take a knotted arc in ordinary space with both ends on the plane \(x = 0\) and the rest of it on the side \(x > 0\), and turn it through \(w\) about that plane. Just like the semicircle, each point sweeps out a circle and the ends stay put, so the arc sweeps out a sphere. The sphere contains a copy of the knotted arc at every angle of the turn.
Artin’s invariant was the fundamental group of the space around the sphere, the group of loops in that space up to continuous deformation. He showed that the spun sphere’s group is the same as the group of the space around the original knot in \(\mathbb{R}^3\). For the trefoil that group isn’t commutative. Around an unknotted sphere it’s the integers, so the spun trefoil can’t be unknotted. Rolfsen’s textbook Knots and Links (1976) covers spinning along with other knotted surfaces.
You can also look at the knotted sphere the way you looked at the bollards, one slice at a time.
At \(w = 0\) the slice is the knotted arc joined end to end with its own mirror image. For the trefoil that’s a right-handed trefoil joined to a left-handed one. Knot theorists call it the square knot, a name borrowed from the reef knot (the square knot in America), whose drawing it resembles. Move ana or kata and the parts of the arc nearest the plane drop out of the slice. The knot breaks into separate loops, and they shrink and vanish at the sphere’s edge. The slices at \(w\) and \(-w\) have the same shape, because the turn is symmetric.
A tarpaulin also rescues the round bollard the rope slipped off. For a sheet, \(p = 2\), so the rule gives \(2 + q = 3\) and \(q = 1\): a closed sheet is held by a bollard whose core is a line.
It’s easiest to see at a single height. In the 3D slice at a fixed height \(z\), the round bollard is a solid ball. A rope loop around a ball just slides off it, but a sheet can wrap the ball completely. Shrink the ball to its core and you have the \(\mathbb{R}^3\) case of the rule, \(2 + 0 = 3 - 1\), a closed sheet held by a point. In ordinary space that’s a balloon with a speck of dust inside. Nobody would call it mooring, but it meets the definition used all along, since the two can’t be separated without one passing through the other. Here the point is one slice of the bollard’s core, which carries on up and down, so the sheet can’t slip off in \(z\) either.
So gather the tarpaulin around the bollard like a sack and fuse the neck shut, the way the harbourmaster spliced the rope. Don’t knot the neck. Gathered up, it’s a thin strand of sheet, and a thin strand tied like rope slips ana just as the bowline did.
4. Casting off
Here is everything in one table. A \(p\)-sphere in \(\mathbb{R}^n\) is held by a core of dimension \(n - 1 - p\). In the table, knots hold only where the codimension is 2.
| A rope loop is held by a core that is | A closed sheet is held by a core that is | Knots in rope loops | Knots in closed sheets | |
|---|---|---|---|---|
| \(\mathbb{R}^3\) | a line | a point | hold | don’t |
| \(\mathbb{R}^4\) | a plane | a line | don’t | hold |
One entry hasn’t come up yet. A sheet in \(\mathbb{R}^3\) has codimension 1, and J. W. Alexander proved in 1924 that every smooth 2-sphere in \(\mathbb{R}^3\) bounds a solid ball, so it can be shrunk to a point and can’t be knotted. The table needs the restriction to spheres. A torus can be knotted in \(\mathbb{R}^3\), as the skin of a thickened trefoil.
Codimension 2 isn’t the only place knots can hold, though. In 1962 André Haefliger found smooth 3-spheres in \(\mathbb{R}^6\) that are knotted, in codimension 3.
Two assumptions did a lot of work. First, “held” meant held by topology alone. Real rope also depends on friction and stiffness, and a 4D rope would be a tube thick in three directions, whose mechanics nothing here describes. Second, every bollard went on forever. That’s the same fact as the splice. The count that stops a loop coming off also stops it going on, so the rope had to be spliced around the bollard and the sack closed around it. A real bollard has a top, and a flared top, the sideways pull of the boat and friction keep the rope on. A real 4D bollard would need the same, flaring out ana and kata as well as up, since the long bollard can’t go on forever in \(w\) either.
Knotted surfaces in four dimensions are still an active area of research.
Fair winds, sailor, and keep a tarpaulin aboard. Next time you’re in, ask the harbourmaster about chains.
References
- Alexander, J. W. (1924). On the subdivision of 3-space by a polyhedron. Proceedings of the National Academy of Sciences, 10(1), 6–8. doi:10.1073/pnas.10.1.6
- Artin, E. (1925). Zur Isotopie zweidimensionaler Flächen im \(\mathbb{R}_4\). Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 4, 174–177. doi:10.1007/BF02950724
- Haefliger, A. (1962). Knotted \((4k - 1)\)-spheres in \(6k\)-space. Annals of Mathematics, 75(3), 452–466. doi:10.2307/1970208
- Hinton, C. H. (1888). A New Era of Thought. Swan Sonnenschein.
- Rolfsen, D. (1976). Knots and Links. Publish or Perish.