How many changes undo a knot

Every knot diagram can be turned into a diagram of the unknot by changing some of its crossings — making the strand that went over go under instead. Choose the crossings so that, walking along the string from some starting point, every crossing is first met on the upper strand. The string then descends steadily, like a spiral staircase, and can be lifted apart. So changing crossings always works, and the only question is how many changes are needed.

The fewest over all diagrams of the knot is its unknotting number. It is one of the oldest and plainest measures of how knotted a knot is, and it is notoriously hard to compute. An upper bound is easy: exhibit some changes that work. A lower bound needs a proof that no cleverer sequence exists, in any diagram, with any amount of rearranging between the changes — and nothing about a single picture can say that.

Half the crossings always suffice #

The staircase argument gives more than existence. Walk along the string and make every crossing an over-crossing when first met, and the diagram descends; make every crossing an under-crossing when first met, and it ascends, which unknots it just as well. The first plan changes some set of crossings and the second changes exactly the others, because each crossing is first met on the upper strand in one plan and on the lower in the other. Together the two plans change every crossing once, so one of them changes at most half.

So a knot drawn with $c$ crossings has unknotting number at most $c/2$. It is a crude bound — it ignores everything about the knot except the size of one diagram — and yet for the two-strand torus knots it is exactly right, since $(c-1)/2$ is the whole number just below $c/2$. For most knots the truth is far smaller: six of the ten knots with six or seven crossings need a single change, and none of them except the seven-crossing torus knot needs as many as three. The bound says how bad a diagram can be; it says nothing about how good the knot is, and a lower bound has to come from somewhere else entirely.

A number each change can move only a little #

The way to prove a lower bound is to find a quantity that the unknot has one value of, the knot has another, and a single crossing change can alter by only a bounded amount. Then the number of changes is at least the distance divided by the bound.

The quantity that works for most small knots is the signature. It was introduced by Trotter in 1962 and its use for unknotting by Murasugi in 1965, and it is a single even integer attached to each knot. The unknot has signature $0$. Changing one crossing changes the signature by $0$ or $\pm 2$. So a knot with signature $\sigma$ needs at least $|\sigma|/2$ changes, and any knotted knot needs at least one.

The picture of the two-strand torus knots is the whole method at its cleanest. The knot with seven crossings, drawn as seven identical twists of two strands, has signature $-6$. Change one crossing and it cancels against the next, leaving five twists: the cinquefoil, signature $-4$. Change another and the trefoil appears, signature $-2$; one more and a single twist remains, which is the unknot. Three changes suffice. Since each change can move the signature by at most two, three are also necessary, and the unknotting number is exactly three.

Every two-strand torus knot goes the same way. The one with $c$ crossings has signature $1 - c$ and is undone by $(c-1)/2$ changes, and the two bounds meet. For torus knots on more strands the analogous statement — Milnor’s conjecture, that the knot of type $(p, q)$ needs $(p-1)(q-1)/2$ changes — is true, but its proof needed gauge theory and arrived in 1993, from Kronheimer and Mrowka. The signature alone is too weak for most of those knots, and the two-strand family is where it happens to be exactly strong enough.

Where the number comes from #

The signature is the signature of a symmetric matrix: the number of positive eigenvalues less the number of negative ones. By Sylvester’s law of inertia that count does not depend on the basis, and a symmetric matrix always has real eigenvalues to count. The work is in choosing a matrix that belongs to the knot rather than to the picture.

The original choice comes from the surface a knot bounds. Take a Seifert surface, choose loops on it that generate its holes, and record how each loop links with a slightly lifted copy of each other one. That gives the Seifert matrix $V$; the signature is the signature of $V + V^{\mathsf{T}}$, and it does not depend on the surface or the loops chosen. That is the definition, and it is not the computation used here.

Gordon and Litherland found in 1978 that the same number can be read from a checkerboard colouring of any diagram, with no surface at all. Shade the regions of the diagram like a chessboard. Build a matrix with a row for each white region: off the diagonal, minus the signed count of crossings where two white regions meet; on the diagonal, whatever makes each row sum to zero. Delete one row and column. This is the Goeritz matrix, and its signature, corrected by a count over the crossings of one orientation type, is the knot’s signature.

The knots here are all drawn as closed braids — which every knot can be, by a theorem of Alexander from 1923 — as strands running down in parallel columns, each crossing a twist between two neighbouring strands, and the ends joined round the side. That is a convenience with a purpose. A braid is also the picture in which the crossings are most visibly a sequence of swaps, the same strings-between-pegs drawing whose crossings count a permutation’s parity, here with the extra information of which strand goes over. In a closed braid the regions can be listed without any geometry — the space between each pair of neighbouring strands is cut into pieces by the crossings in that column — so the Goeritz matrix is a short computation, and nothing depends on reading a picture correctly.

Every braid word is checked against things it did not produce. The determinant of the Goeritz matrix is the knot’s determinant, the number that counts its colourings, and it must match the tables: 3 for the trefoil, 15 for $7_4$, and a different value for each of the seven-crossing knots, so a wrong braid would be caught. Both checkerboard colourings must give the same signature. The signature must be even. And it must satisfy Levine’s congruence — divisible by four exactly when the determinant is one more than a multiple of four — which ties it to the determinant by an argument that has nothing to do with Goeritz. The determinant is the value at $-1$ of the Alexander polynomial, and the congruence is really a statement about the sign of that value, which the signature fixes.

None of these checks is a proof that the signature is an invariant — that it is the same for every diagram of the knot. That is a theorem, proved by showing the corrected Goeritz signature survives each of Reidemeister’s three moves. The figures take it as given and use the checks to confirm they are computing it, not to confirm it exists.

One change, at most two #

The claim that one crossing change moves the signature by at most two is what makes it useful, and it is worth seeing why it is true. Choose a Seifert surface on which the crossing to be changed sits in a single twisted band. Changing the crossing adds a full twist to that band, and the only entry of the Seifert matrix that notices is the one recording how the loop through that band links with its own lifted copy — it moves by one. So $V + V^{\mathsf{T}}$ changes by two in a single diagonal entry.

A symmetric matrix changed in one entry of its diagonal is changed by a matrix of rank one, and a rank-one change can move each eigenvalue only past its neighbour, not beyond — the same interlacing that governs where a symmetric matrix’s eigenvalues sit. So at most one eigenvalue changes sign, and the signature, which counts signs, moves by at most two. The direction is fixed as well: the change adds or subtracts a positive quantity depending on whether the crossing was positive or negative, and that is why every positive crossing in the census moved the signature up or not at all.

The census checks this over every single crossing change in the fourteen braids, ninety-two of them. It is evidence the theorem is being computed correctly rather than evidence for the theorem, which needs no help; but a mistake in the Goeritz computation would almost certainly have produced a jump of four somewhere, and none appeared.

Thirteen knots settled, one not #

The table runs the method on every prime knot with seven or fewer crossings. The lower bound is the larger of $|\sigma|/2$ and one. The upper bound is found by searching every set of crossings in the braid word, from the smallest up, for one whose change leaves the unknot. Recognising the unknot uses the Jones polynomial: the changed braid is accepted only if its polynomial is exactly $1$, and no diagram with this few crossings has polynomial $1$ unless it is unknotted.

For thirteen knots the two bounds meet, and the unknotting number is settled by a matrix and a search. Four of them — the figure-eight knot, $6_1$, $6_3$ and $7_7$ — have signature zero, and for those the lower bound is only the trivial one: they are knotted, so at least one change. Since one change is found, one is the answer, and the signature has contributed nothing but has not needed to.

The knot $5_2$ is the typical settled case. Its signature is $-2$, which forces at least one change, and the search finds that changing the first crossing of its braid leaves the unknot. The whole argument fits in one figure: the knot, the ringed crossing, the unknot it becomes, and a number that says no fewer changes could work.

The fourteenth knot is $7_4$, and it is where the signature runs out. Its signature is $-2$, so at least one change is needed. The search finds no single crossing of its braid whose change unknots it, and finds a pair that does. But a search in one diagram proves nothing about others: perhaps some other diagram of $7_4$ has one crossing whose change would do it. The signature cannot rule that out, and for a long time nothing could. Lickorish settled it in 1985, using the linking form of a three-dimensional space built from the knot — the double cover of space branched along it — which carries finer information than the signature. The unknotting number of $7_4$ is two.

What the signature sees and what it does not #

It detects handedness. Reflecting a knot negates its signature, since every crossing changes sign. A knot with non-zero signature is therefore not the same as its mirror image: the trefoil has signature $-2$ and its reflection $+2$. That reproves, with a matrix, what the bracket polynomial showed about the trefoil — and like the bracket, it is silent on knots that happen to have signature zero, whether or not they are their own mirror images.

It adds. Tie two knots one after the other in the same string and the signatures add. So the seven-crossing torus knot tied beside its mirror image has signature $-6 + 6 = 0$, and the signature says nothing at all about how many changes that composite needs. It had long been expected that unknotting numbers add too, so that the composite would need three plus three. In 2025 Brittenham and Hermiller reported a way to undo it in five, which shows that unknotting number is not additive — a question that had been open for decades, answered with an explicit sequence of changes.

It also bounds a surface in four dimensions. Murasugi’s argument gives more than a bound on crossing changes: half the absolute signature is at most the genus of any surface the knot bounds in the four-dimensional ball, the slice genus. The seven-crossing torus knot cannot bound a disc in the four-ball, because its signature is not zero. That is a first tool for the question of which knots are slice, and it is far from the last. A surface in ordinary space can be pushed into the four-ball, so half the signature is also at most the ordinary genus; for the two-strand torus knots, whose Seifert surfaces have $(c-1)/2$ handles, the signature bound, the four-dimensional genus, the ordinary genus and the unknotting number all come out equal, which is part of why that family is so often the first example of anything.

What the pictures cannot show #

The search is in one diagram, and the unknotting number is about all of them. Each ringed pair in a figure is an upper bound found in a braid diagram. Unknotting number allows any diagram and any rearranging between changes, so a smaller number found in a different diagram would override the search. For thirteen knots the signature closes that door; for $7_4$ the figure shows two changes and cannot, on its own, show that one is impossible.

The Jones polynomial is used as a recogniser, not a proof of unknotting in general. Whether it detects the unknot is open. Here it is used within its proven range: at so few crossings the only knot with polynomial $1$ is the unknot, because the tables of small knots are complete and every other entry has a different polynomial.

The signature’s definition from a surface is described, not drawn. The Goeritz computation is what the figures run, and Gordon and Litherland’s theorem is what makes it equal to the surface definition. The two agree on every knot here, but the figures only compute one of them.

Still open: knots nobody can undo on paper #

The unknotting number is known for every prime knot of up to nine crossings — the last few of them settled only in the 2000s — but several knots with ten crossings still have no settled value: for each, some diagram gives an upper bound and every known invariant gives a lower bound one short of it. The signature is one of those invariants, and its descendants from the 1990s and after — bounds from gauge theory and from Floer homologies — have settled many of the cases it could not. What remains are knots where the gap between exhibited changes and provable necessity is exactly one, and no method yet closes it.

The other open direction is structural. That unknotting numbers do not always add is now known; how badly they can fail to add, and which invariant would predict when a composite knot is easier than its pieces, are not.

A distance measured from below #

The unknotting number is a distance from a knot to the unknot, counted in crossing changes. Upper bounds are paths: a sequence of changes, shown. Lower bounds are fences: a quantity with a Lipschitz constant, so that no path can cross the distance faster than it allows. The signature is the simplest fence with a constant of two, and it is built from nothing more than a symmetric matrix of crossing counts.

Where the fence and the path meet, the distance is known exactly, and where they do not, it is known only up to the gap. For the small knots the gap is almost always zero, which is why a table of them can be filled in by a matrix and a search. The knot $7_4$ is the first place where a fence of a different kind was needed, and it shows, in seven crossings, why a measurement from below is always the harder half.