The Unknotting Number Is Not Additive

Summary

On June 30, 2025, Mark Brittenham and Susan Hermiller uploaded a preprint to the arXiv called “Unknotting number is not additive under connected sum” (and an updated version on September 15, 2025). In it, they surprised the mathematical community by giving a counterexample to a long-standing conjecture in knot theory. The story was picked up by publications like Scientific American and Quanta and by math YouTuber Matt Parker. The conjecture is easy to understand, although we need a few definitions first. (Mathematical) knot: We can think of a mathematical knot as a loop of string sitting in three-dimensional space. In other words, if we took a piece of string, tied a knot, and then glued the two ends together, we’d get a mathematical knot. Knot projection: Given any mathematical knot, we draw a two-dimensional version of it in the plane. It is like the shadow of the knot, but with breaks in the knot to indicate which strand is on top and which is on the bottom. Unknotting number: If we have the projection of a knot, we can change some crossings (change which strand is over and which is under) to make it unknotted (called the unknot). To compute the unknotting number of a knot K, u(K), we look at all possible projections and find the fewest number of crossing changes we must make to obtain the unknot. Connected sum: Given two knots, J and K, we can cut each knot at one point and join the cut ends to form a new, larger knot, J#K. This is called the connected sum of the knots. Below, we see a knot called the (2,7) torus knot and its mirror image (left), and their connected sum on the right. Although unknotting numbers are notoriously difficult to compute, we know that the unknotting number of a (p,q) torus knot is (p-1)(q-1)/2. So, the (2,7) torus knots above have unknotting number (2 – 1)(7 – 1)/2 = 3. It is not difficult to check that by changing three crossings of the projections shown above, the (2,7) torus knot becomes the unknot. It turns out that changing two cr...