Zero can mean two different things

Two loops and one number got the linking number by counting crossings in a picture: give each place one strand passes over the other a sign, add the signs, halve the total. That is a recipe, and a recipe is not a meaning. It says what to do and leaves open why the answer should be worth anything.

The meaning is in the picture above. Take one of the two loops and fill it in: stretch a surface across it, any surface at all, with the loop as its rim. Now count how many times the other loop passes through that surface, giving a puncture a plus when it goes through one way and a minus when it goes the other. That count is the linking number.

The count, and why the surface does not matter #

The definition looks fragile. There are infinitely many surfaces spanning a given loop — a flat disc, a bowl, a long drooping bag — and the second loop hits different ones in different places. What saves it is that any two spanning surfaces, glued along their common rim, make a closed surface, and a closed loop crosses a closed surface an equal number of times in each direction. So the difference between the two counts is zero, and the counts agree.

That argument is worth pausing on, because it is the shape of a great many arguments in topology: a quantity that appears to depend on a choice is shown not to, by observing that two choices differ by something closed, and that closed things contribute nothing. The linking number is the smallest example anybody meets.

Once the count is trusted, the recipe from the rung below falls out of it. Push the second loop until it lies almost flat against the disc; every crossing of the diagram is then a place where the loop dips through and comes back, or passes over without going through. The dips are the punctures, and the signs of the crossings are the directions of the dips. Halving the crossing sum is bookkeeping to avoid counting each dip twice.

Two punctures, opposite ways #

Here is the thing the count makes visible and the crossing recipe hides. Zero is the answer for two entirely different reasons.

A loop that never goes near the disc contributes nothing because there is nothing to contribute. A loop that goes through and comes back contributes nothing because two contributions cancel. Arithmetically these are the same; topologically they need not be.

For the loop drawn in the hero figure, the two readings happen to agree: the small loop really can be slid sideways off the disc and carried away, and once it is clear of the disc it never went through at all. But nothing in the number said so. The number was zero in both cases and the pulling-apart was established by looking.

The link where looking is not enough #

The famous case is due to J. H. C. Whitehead, and it is the reason this rung exists.

Take two loops. The first is an ordinary round circle. The second is arranged to go through the first’s disc twice in opposite directions — so the punctures cancel and the linking number is zero — but arranged so that the two passes are clasped: before the second loop turns round to come back, it hooks through itself. The result is the Whitehead link, and it cannot be pulled apart.

That last sentence is a claim no figure on this page proves, and it should be read as the claim it is. The pictures here establish what the linking number is and what it counts. They do not establish that a particular pair of loops is inseparable, and no picture could: separating two loops means exhibiting a deformation, and failing to separate them means ruling out every deformation, which is an infinite task and not a drawing.

What rules them out is a different kind of object. The linking number is what topologists call a first-order invariant: it sees how one loop sits in the homology of the other’s complement — roughly, how it counts against surfaces — and homology cannot tell a clasp from nothing, because a clasp adds one puncture and subtracts another. The Whitehead link is detected by finer machinery: Milnor’s invariants, which look at how the loops sit in the fundamental group of the complement rather than in its homology, and which for this link give a value of one where the linking number gives nothing.

There is an honest way to state what the picture does show. The linking number rules links out and never rules them in. A non-zero value proves that two loops cannot be separated, because separating them would have to change an integer continuously. A zero value proves nothing whatever.

Three rings and no pair #

The other standard demonstration needs no clasp and does not have to be taken on trust in the same way, because it can be drawn.

Any two of the three rings, taken alone, are unlinked in the strongest sense: lift one away and it comes free. All three together do not come apart. So the pairwise linking numbers, which are all zero, are not merely uninformative about the triple; they are correct about every pair and silent about the arrangement.

That is the sharper way to say what is going on. The linking number is a function of two loops. A property of three loops that is invisible in every pair is not something a function of two loops can be expected to see, and expecting it to is a category error rather than a defect in the number.

The same construction, one dimension at a time #

The puncture count is worth stating in its general form, because the generality is what makes it feel inevitable rather than clever.

Two objects sitting inside a space can be compared whenever their dimensions add up to one less than the space’s. A point and a line in the plane: zero plus one is one, which is one less than two, and the comparison is does the line pass through the point. Two curves in three dimensions: one plus one is two, one less than three, and the comparison is the linking number. Two surfaces in five dimensions, a curve and a surface in four — the same arithmetic, the same construction, and the same theorem that the answer does not depend on which spanning object was chosen.

Seen that way the linking number is not a fact about knots at all. It is a fact about how a boundary meets a cycle, and knots are where it happens to be visible. The signed crossing recipe from the rung below is a two-dimensional shadow of the intersection, which is why halving is needed: a projection double-counts.

The nearest relative on this site is the crossing count that decides whether a point is inside a closed curve. Which side of the line is inside fires a ray from a point and counts how often it meets the curve; the parity of that count answers the question, and the answer does not depend on the ray. That is the linking number’s argument with one dimension removed from everything, and the reason both are stable is identical — a ray moved continuously gains and loses crossings in pairs.

The same construction one dimension up is what nothing on a sphere can be combed flat is about, where the count is of zeros of a vector field rather than of punctures, and the total is forced by the shape of the space rather than by the field.

What the number cannot be talked out of #

Against all of that, the linking number is remarkably hard to break, and the second half of this rung is about how hard.

Deform the loops however seems good: the crossings of the diagram appear and disappear in pairs, and a pair that appears has opposite signs, so the sum is untouched. Change the direction of view and the diagram is redrawn from scratch, with different crossings in different places, and the sum is untouched. Compute Gauss’s double integral instead — an analytic object with no diagram anywhere in it — and the same integer comes out.

The reason all three agree is the puncture count. Every one of them is measuring the same thing: how many times, with signs, one loop crosses a surface the other bounds. A deformation moves the surface; it does not change the algebra of crossing it. That is why the number is so stable, and it is also precisely why it is blind to the clasp: a clasp does not change how often anything crosses anything.

Where it fails, and what it needs #

Both loops have to be oriented. Reverse the direction of travel along one of them and every puncture changes sign, so the linking number changes sign. This is not a flaw — an unoriented pair has a linking number only up to sign, and the absolute value is the invariant then. But a figure that draws two loops without arrows on them has not quite drawn the thing the number is about.

The loops have to be disjoint. If they touch, there is no surface to count punctures of, and the number is undefined rather than zero. This matters more than it sounds: a great deal of the work in defining invariants of one loop — the self-linking number, the framing — is the work of pushing a loop off itself in a chosen way so that the definition applies at all.

It is an integer, so it cannot be a little bit wrong. That is the source of all its strength, and the source of its limitation. Anything a homotopy can do continuously, an integer cannot follow; so the number is preserved. Anything a homotopy cannot do is invisible to it for the same reason.

And the whole thing needs a spanning surface to exist. For a loop in ordinary space one always does, but the definition as stated does not survive being moved to a space where some loops bound nothing — in a solid torus, the core circle bounds no surface at all, and the linking number has to be redefined by a different route or given up.

There is a related trap in the choice of surface. Nothing above required the spanning surface to be a disc, or even to be orientable; but the signs did require an orientation, since a puncture is a plus or a minus according to which side of the surface the loop arrives from. Span the loop with a Möbius band and there are no two sides to arrive from, so there are no signs, and the count that survives is a count modulo two rather than an integer. That weaker invariant is real and useful, and it is not the linking number.

Where it came from #

Gauss wrote the double integral in a notebook in 1833, in four lines, with no proof and no comment except that it belonged to the geometria situs — the geometry of position — which he thought had barely begun. His interest was physical: the integral is the work done carrying a magnetic pole round one loop while a current flows in the other, and its being a whole number is the statement that the work comes in fixed packets.

The count-the-punctures reading came much later and from a different direction, with the algebraic topology of the 1920s and 1930s. It is the version that generalises: replace the loops by cycles of any dimensions in any space, replace the disc by a chain the first bounds, and the intersection number is exactly the same construction. The linking number of two curves in three dimensions is the smallest interesting case of an intersection theory that runs all the way up.

The Whitehead link arrived in 1934, in a paper correcting a mistake. Whitehead had earlier believed a certain three-dimensional space was a sphere; he showed it was not, by building a link whose complement did the work, and the link is the part everybody remembers. It is the standing example of the gap between homology and homotopy, and it is why nobody in the subject says “the linking number is zero, so they come apart”.

What the pictures cannot show #

The disc drawn here is flat and the second loop is small, and both are conveniences. The surface may be any shape at all — the count is the same for a bowl, a bag or a thing with a long neck — and the figure draws the easiest one rather than the general one, which is a claim about the theorem the figure does not make.

The punctures are marked at points where the drawn curve crosses the plane the disc lies in, which is exactly right for a flat disc and would need restating for a curved one. A reader who takes the marked points to be the definition will be surprised by the first spanning surface that is not flat.

And, most importantly, no figure here shows that a pair of loops cannot be separated. Two of the links drawn come to zero, and the essay says which one comes apart and which one does not; that difference is stated on the authority of results the pictures do not contain. A figure can perform a search and report what it did not find, and no figure on this page performs the relevant search, because the relevant search is over an infinite set of deformations.

The ladder from here #

Below: two loops and one number, which is where the count comes from, and the same loop, unrolled, whose lifting argument is the same idea with the surface replaced by a covering. Sideways: three moves and what they cannot undo, where the invariance argument is run for a single knot instead of a pair, and colours that count more than three, which is an invariant that sees exactly what this one does not. Above: Milnor’s invariants, the Alexander polynomial of a link complement, and the self-linking number a framing produces.

What is worth carrying away #

An invariant is a promise in one direction only. When it differs, the objects differ; when it agrees, nothing has been said. That asymmetry is easy to state and remarkably easy to forget, because a number that is right about everything it detects starts to feel like a number that detects everything.

The puncture count is the reason to expect the gap rather than be surprised by it. It measures a loop against surfaces, and surfaces are a coarse instrument: they cannot tell a hook from a straight pass, because both cross a surface the same number of times. What the linking number sees is exactly what a count of crossings can see, and the Whitehead link is the smallest thing that lives in the difference.