Marc Lackenby

Marc Lackenby: The Man Who Untangled the Unknot

"No systematic method is yet known by which one can tell whether two knots are the same." That was Alan Turing in 1954, flagging one of the most stubbornly concrete unsolved problems in mathematics: given a tangle of string with its ends joined, decide whether it is genuinely knotted. Max Dehn had posed it in 1910. Wolfgang Haken produced the first algorithm in 1961, but it was hopelessly slow on anything complicated. On 2 February 2021, at UC Davis, Marc Lackenby announced that the question could be settled in *n* raised to the power *c* log *n* steps — quasi-polynomial time, a whisker off the efficiency that computer scientists regard as the mark of a genuinely tractable problem. Oxford's mathematicians called it a Gordian tour-de-force.

Cambridge and the Lickorish School

Lackenby went up to Cambridge to read mathematics in 1990 and stayed for his doctorate, completed in 1997 under W. B. R. Lickorish with a thesis titled "Dehn Surgery and Unknotting Operations." Lickorish is one of the central figures in twentieth-century knot theory, and the topics of the thesis — Dehn surgery and unknotting — would still be recognisably Lackenby's subjects a quarter of a century later. Few mathematicians pick their territory that early and then keep finding new things in it.

He held a Miller Research Fellowship at UC Berkeley and a research fellowship back at Cambridge before moving to Oxford in 1999 as a lecturer and Fellow of St Catherine's College. Promotion to a professorship followed in 2006. His work sits across knot theory, low-dimensional topology and group theory — three subjects that in three dimensions turn out to be more or less the same subject seen from different angles.

Filling in the Holes

Dehn surgery is the operation at the centre of three-dimensional topology: cut out a solid tube around a knot, then glue it back differently. Almost every three-manifold can be produced this way, so understanding the operation means understanding the objects.

The question that matters is when the result stays hyperbolic — hyperbolic geometry being the structure that makes a three-manifold rigid and analysable, and its absence being where the wilderness begins. The 2π theorem gives a geometric criterion guaranteeing that surgery preserves hyperbolicity. Lackenby proved a strengthened version of it. That is a characteristic kind of result for him: not the discovery of a new landscape but a sharpening of an existing boundary line, done well enough that a great deal more territory falls on the useful side of it.

Alternating Knots and Volume

A second strand connects a knot's picture to its geometry. Draw a knot as a diagram on a page and you have combinatorics — crossings, arcs, regions. Consider the space around the knot and you have geometry, with a hyperbolic volume attached to it. These are utterly different kinds of information, and relating them is one of the deep projects of the field.

Lackenby produced bounds on the hyperbolic volume of knot complements for alternating knots — the class in which crossings alternate over, under, over, under as you travel around — tying a geometric invariant to something you can literally count on a drawing. Results of this kind are what make knot geometry computable rather than merely contemplable.

Reidemeister's Moves, Counted

Any diagram of a knot can be turned into any other diagram of the same knot by a sequence of three elementary manipulations, the Reidemeister moves. That has been known since the 1920s, and it is useless on its own: nothing tells you how many moves are needed, and the sequence may have to pass through diagrams far messier than either endpoint before it simplifies. That is exactly why unknotting is hard — you often have to make a tangle worse before it can get better.

Lackenby showed that a diagram of the unknot can be untangled in a number of Reidemeister moves polynomial in the number of crossings. The pessimism was misplaced. There is always a reasonably short route to the trivial circle, even if finding it is another matter.

Quasi-Polynomial Time

Which brings us to 2021. The unknot recognition problem had accumulated many algorithms after Haken's, drawing on topology and on geometry, and nearly all of them were catastrophically slow once knots got complicated. Whether a polynomial-time algorithm existed had become one of the famous open questions at the border of topology and computer science. William Thurston, in 2011, put it plainly: "A lot of people have thought about this question ... but this has been a very hard question to resolve."

Lackenby's *n*^(*c* log *n*) bound does not quite settle it. Quasi-polynomial is slower than polynomial. But the gap between quasi-polynomial and everything that came before is enormous, and the gap between quasi-polynomial and polynomial is, in practical and conceptual terms, small. A problem Turing had listed among the untamed is now, essentially, tamed.

Why Marc Is Called a Genius

The quality on display is not speed and not breadth. It is a kind of patient depth: staying with the same cluster of questions — Dehn surgery, unknotting, the geometry of knot complements — for twenty-five years, accumulating technique until problems that had resisted generations became reachable. The unknot result is the clearest evidence, because it did not fall to a clever trick. It fell to somebody who had spent a career building the specific machinery required and could see, where others could not, which combination of topological and geometric tools would compose.

The recognition from his peers has been consistent rather than sensational: the London Mathematical Society's Whitehead Prize in 2003, a Philip Leverhulme Prize in mathematics and statistics in 2006, an invitation to speak at the International Congress of Mathematicians in 2010 — the discipline's most reliable signal that a body of work matters. The 2021 announcement drew the strongest language, with Oxford reaching for "tour-de-force."

The counter-case has to be stated. No source records anyone calling Lackenby a genius, and the mathematical culture he belongs to is notably sparing with the word. His work builds directly on Haken, Thurston, Lickorish and the whole normal-surface tradition; the quasi-polynomial algorithm is a spectacular improvement within an established framework, not a new framework. It is also not polynomial, which means the headline question is still technically open. And low-dimensional topology, for all its beauty, is a specialist field — his results are famous among topologists and largely unknown outside. What is beyond dispute is that he cracked something that had held out for a century, and did it by knowing his own subject more thoroughly than anyone else.

Legacy

What Lackenby has done, across all four strands of his work, is to make knot theory *effective*. Bounding volumes by counting crossings, bounding untangling by counting moves, bounding recognition by counting steps — in each case an object that could be contemplated became an object that can be computed. That shift is what turns a beautiful theory into a usable one, and it is the reason the 2021 result travelled well beyond topology into theoretical computer science. Dehn asked the question in 1910. It took 111 years and a professor at Oxford who refused to work on anything else.

Achievements

• Whitehead Prize — 2003
• Held posts at University of Oxford