The surface a knot bounds

A loop of wire dipped in soapy water comes out holding a film. If the loop is a plain circle the film is a disc. If the wire is knotted the film is still there, stretched across whatever the knot allows, and it is a surface whose edge is the knot.

That observation turns a question about string into a question about surfaces, and surfaces are completely classified. A surface with one edge is a sphere with some handles and a hole, and the number of handles is its genus. So every knot is the edge of surfaces of various genera, and the smallest genus it can manage is a number attached to the knot itself: the genus of the knot. The unknot bounds a disc, which has no handles, and a knot has genus nought exactly when it is the unknot. That makes the genus an invariant with a use no crossing count has: a surface-theoretic certificate that a knot is knotted, as soon as it is shown that no disc will do.

The difficulty is that a soap film is not a proof. For a knotted wire the film may be one-sided, it may be pinched, and it is certainly not obviously the surface with fewest handles. What is needed is a construction that produces a surface with two sides from any diagram, and a way to count its handles.

Smoothing every crossing the way the arrows allow #

Herbert Seifert published the construction in 1934, and it begins by choosing a direction along the knot.

At each crossing two strands pass, one over the other, each with an arrow. There are two ways to remove the crossing by cutting both strands and reconnecting the four loose ends, and exactly one of them keeps the arrows consistent: the incoming end of each strand is joined to the outgoing end of the other. Do that at every crossing. What is left has no crossings at all, so it is a collection of disjoint closed curves in the plane — the Seifert circles — each with a direction.

For the trefoil the smoothing leaves two circles, one nested inside the other. For the figure-eight above it leaves three: an outer circle and two inner ones. For the cinquefoil, and for every knot that winds twice round a torus, it leaves exactly two.

The circles may be nested, and the construction handles nesting by height. Fill each circle with a flat disc, and where one circle sits inside another, lift the inner disc slightly above the outer so the two do not meet. The result is a stack of disjoint discs, each with an oriented edge.

A band for every crossing #

The crossings have been removed, and they now have to be put back. At each one the two smoothed strands belong to two different circles, lying side by side or one above the other, and the original crossing is restored by joining those two discs with a narrow band carrying a single half twist, twisted in the direction the crossing went.

The edge of the discs-and-bands surface is the original knot: along each disc’s rim it follows a Seifert circle, and at each band it crosses from one disc to the next exactly as the original strands crossed. And the surface has two sides. Colour the side of each disc that the circle’s arrow runs anticlockwise round as the top; the half twist in each band is exactly what is needed for the top of one disc to meet the top of the next, because the two circles run in opposite senses where the band joins them. That consistency is the whole reason for smoothing by the arrows — the other smoothing would give a surface that could be one-sided, like the Möbius band the trefoil also bounds. Orientation is a sign chosen consistently across a surface, and the arrows on the knot are what choose it here.

Counting the handles without drawing them #

The surface’s genus can be read off from two numbers the diagram already has: the number of crossings $c$ and the number of Seifert circles $s$.

A disc has Euler characteristic one. Attaching a band along two short arcs of the boundary adds one face and two edges’ worth of gluing, and lowers the characteristic by one. So $s$ discs and $c$ bands give

$$\chi = s - c.$$

A connected surface with genus $g$ and $b$ boundary circles has $\chi = 2 - 2g - b$ — the third number a surface needs is $b$, and a knot has one. Setting $b = 1$ and solving:

$$g = \frac{c - s + 1}{2}.$$

For the trefoil that is $(3 - 2 + 1)/2 = 1$: the surface is a torus with a hole punched in it, and the trefoil is its edge. For the figure-eight it is $(4 - 3 + 1)/2 = 1$ as well.

The discs-and-bands drawing is a graph — a vertex for each disc, an edge for each band — and the count has a second reading in those terms. The graph is connected, since the knot is one piece, and its number of independent cycles is $c - s + 1$. So twice the genus is the number of independent loops in the graph of circles and crossings. Each independent loop of bands contributes half a handle, and two of them make a full one.

The cinquefoil’s surface is two discs joined by five parallel bands. The pattern for the knots that wind twice round a torus is visible from it at once: a $(2, q)$ torus knot always smooths into two circles, its $q$ crossings are $q$ parallel bands, and its surface has genus $(q - 1)/2$ — one for the trefoil, two for the cinquefoil, three for the seven-crossing knot.

An upper bound, from any diagram #

Seifert’s algorithm builds a surface, and the knot’s genus is the fewest handles on any surface it bounds. So the count gives an upper bound: the genus of a knot is at most $(c - s + 1)/2$ for every diagram of it.

The bound depends on the diagram. Add a twist to a strand with a Reidemeister move and the diagram gains a crossing; the smoothing gains a small circle; $c - s$ is unchanged and the bound survives. Other moves can change $c - s$, and a clumsy diagram of the unknot can produce a surface with many handles. Taking the best diagram helps, but even that is not enough in general: there are knots whose minimal surfaces are not produced by Seifert’s algorithm on any diagram. The algorithm’s best is called the canonical genus, and it can exceed the genus.

An upper bound is not an invariant anybody can compare. What makes the genus computable in good cases is a matching lower bound, and it comes from the polynomial.

A lower bound, from the polynomial #

The Alexander polynomial was computed from a matrix with a row per crossing. Seifert showed that it can also be computed from any surface the knot bounds.

A surface of genus $g$ with one edge has $2g$ independent loops on it. Push each loop slightly off the surface to its top side, and record how many times it links each of the original loops. That gives a $2g$-by-$2g$ matrix of whole numbers, the Seifert matrix $V$. The polynomial is

$$\Delta(t) \;\doteq\; \det\big(V - tV^{\mathsf T}\big),$$

up to the familiar factor $\pm t^k$. The determinant of a $2g$-by-$2g$ matrix whose entries are linear in $t$ is a polynomial of degree at most $2g$. So the polynomial’s span — the gap between its highest and lowest powers — is at most $2g$, for every surface the knot bounds, and in particular for the one with fewest handles:

$$\frac{\text{span of } \Delta}{2} \;\le\; g(K) \;\le\; \frac{c - s + 1}{2}.$$

The lower bound is a property of the knot and the upper bound is a property of a diagram, and the knot’s genus is squeezed between them.

Where the two bounds meet #

On every row of the table the two bounds coincide, and that is not luck. All four diagrams are alternating: travelling along the knot, the crossings go over, under, over, under without exception.

Crowell and Murasugi proved independently in 1958 that for a reduced alternating diagram the Alexander polynomial’s span is exactly $c - s + 1$. The matrix minor, for such a diagram, has a leading and a trailing coefficient that can be counted combinatorially from the circles, and neither can cancel. So for every alternating knot the surface Seifert’s algorithm builds from its alternating diagram has the fewest handles possible, and the genus is read off a picture with no search at all.

Most small knots are alternating — every knot with seven crossings or fewer is — which is why the genus of the knots in any table is usually easy. The difficulty begins with the non-alternating knots, where the two bounds can separate as far as they like.

Tying a knot never undoes another #

The genus has one property that makes it indispensable, and it answers a question that sounds as though it should be about string rather than about algebra.

Tie a knot in a string and then tie a second knot further along. The result is the connected sum of the two. Can the second knot ever undo the first — can two genuinely knotted strings combine into something that pulls straight?

Horst Schubert proved in 1949 that genus adds under connected sum: $g(K \# J) = g(K) + g(J)$. One direction is easy — glue minimal surfaces for $K$ and $J$ along an arc of their edges and the handles add. The other direction, that no cleverer surface for the sum can do better, is the real theorem, and it is proved by cutting a minimal surface for the sum along the sphere that separates the two knots.

Given additivity the question is closed at once. The unknot has genus nought and every other knot has genus at least one. If $K \# J$ were the unknot, $g(K) + g(J)$ would be nought, so both would be nought, so both would be unknots. No knot has an inverse. The same additivity shows every knot is a sum of prime knots — ones that are not themselves sums — since each factor removes at least one from the genus and the genus is finite, and Schubert proved the factorisation is unique, exactly as whole numbers factor into primes in one way.

The analogy runs deeper than the word prime. Genus plays the part of the logarithm of a whole number: it adds when knots are multiplied, it is nought only for the unit, and a knot of genus one — like a prime number — cannot be split, since a split would need two factors each of genus at least one. The trefoil and the figure-eight are prime for that reason alone, read straight off the table above. Not every prime knot has genus one, just as not every prime number is small, and deciding primality for a knot of higher genus needs more than counting — but the easiest cases fall to the count, which is exactly how the smallest cases of unique factorisation fall to size.

There is a well-known way to seem to cheat this, and it shows exactly where the theorem’s hypotheses sit. An infinite string with infinitely many trefoils tied in it alternately with their mirror images can be regrouped, pair by pair, into infinitely many cancelling pairs or into one trefoil followed by the same infinite string — an argument known as the Mazur swindle. It proves nothing about finite knots, because a genus cannot be infinite, and it is precisely the counting that breaks.

When the polynomial’s bound is useless #

For a non-alternating knot the polynomial’s bound can be far below the genus, and the extreme cases are the knots whose polynomial is 1.

The Conway knot, with eleven crossings, has Alexander polynomial 1, so the polynomial’s lower bound on its genus is nought — the value for the unknot. Its genus is 3. The Kinoshita–Terasaka knot, its mutant, has polynomial 1 and genus 2. For both, the squeeze collapses and the genus has to be found some other way.

The other way arrived in 2004, when Ozsváth and Szabó showed that knot Floer homology — a much richer invariant whose graded Euler characteristic is the Alexander polynomial — detects the genus exactly. Where the polynomial’s coefficients cancel and the span shrinks, the homology’s ranks cannot cancel, and the highest grading in which it is non-zero is the genus. It is the lower bound made sharp, and it confirms that the polynomial was the shadow of an invariant that had been measuring handles all along.

The genus is also decidable directly, without any invariant. Haken’s theory of normal surfaces from the 1960s gives an algorithm that finds a minimal surface by searching a finite list, and it has since been shown that a claim that a knot has genus at most $g$ can be checked quickly if the right surface is supplied. The search itself is expensive, and deciding the corresponding question for knots in more general three-dimensional spaces is known to be as hard as any problem whose answers can be checked quickly.

What the discs and bands leave out #

The surfaces here are drawn as graphs — discs as circles, bands as strips — and a graph forgets almost everything about how the surface sits in space. It does not show which disc is lifted above which, nor which way each band twists, nor that the bands of the figure-eight’s surface pass on both sides of the middle disc. The count of handles survives that forgetting; the embedding does not, and it is the embedding that makes the edge a particular knot.

The table’s agreement is shown for four alternating knots, and the theorem it illustrates is about all alternating knots. The knots where the bounds disagree cannot be drawn at a size where their smoothings are legible, so the gap between the canonical genus and the genus is described and not exhibited.

And none of the pictures shows a minimal surface being minimal. That the trefoil has no surface without a handle is the statement that it is knotted, and it is proved here by the polynomial’s lower bound, a determinant — not by anything a drawing of a surface could display.

Still open: whether every slice knot is ribbon #

The surfaces above live in ordinary space, and a knot in space can also be regarded as the edge of a surface living in four dimensions — sitting on the boundary sphere of a four-dimensional ball, with the surface allowed to dip into the ball’s interior. There is more room there, and a knot may bound a surface with fewer handles; the fewest is its slice genus, which is at most the genus and can be much smaller. A knot with slice genus nought bounds a disc in the four-ball and is called slice.

Some slice knots are slice for a visible reason. A ribbon knot bounds a disc in ordinary space that is allowed to pass through itself along short segments called ribbon singularities, and any such disc can be pushed into the four-ball to remove the self-intersections. Every ribbon knot is therefore slice. The square knot — a trefoil tied beside its mirror image — is ribbon, and so slice, although its genus is two.

Ralph Fox asked in 1962 whether every slice knot is ribbon, and the question is still open. It is known for the two-bridge knots, a large and well-understood family, and it has been checked on every knot with up to twelve or so crossings for which sliceness is decided. The difficulty is that sliceness is a four-dimensional property, while ribbon discs are built in three dimensions, and the tools for ruling out a disc in four dimensions — gauge theory, Floer homologies — do not construct discs when they fail to rule them out. How hard that can be was shown in 2020, when Lisa Piccirillo settled whether the Conway knot is slice — it is not — a question that had stood for fifty years and had resisted every one of those tools until she found a different knot sharing its four-dimensional behaviour.

A film stretched across a knot #

Seifert’s construction replaces the knot, a curve, with a surface, an object with a classification. That exchange is the source of every result above: the genus is a surface’s handle count, the polynomial’s degree is bounded by the size of a matrix built from loops on the surface, and additivity is proved by cutting surfaces along a sphere.

It is also why the colourings, the determinant and the polynomial fit together. Each was computed from crossings, and each turns out to be a fact about the surface the crossings build — the determinant is $\det(V + V^{\mathsf T})$, the polynomial is $\det(V - tV^{\mathsf T})$, and the genus is half the size of $V$. A knot diagram is a flat record of a surface’s edge, and the flat invariants were reading the surface through its edge the whole time.