The unknotting number of a knot is the fewest crossing changes needed to untie it. Change a crossing — make the strand that goes over go under, or vice versa — and eventually any knot becomes the unknot. For the trefoil, one crossing change suffices. For more complicated knots, you need more.
In 1937, Wendt conjectured that unknotting numbers add. If you tie two knots together — the connected sum — the number of crossing changes needed to unknot the combination should equal the sum of the individual unknotting numbers. This seems obvious: the two knots sit in separate regions of the combined knot, so untying one doesn't help with the other.
Brittenham and Hermiller (arXiv 2506.24088) proved that this is wrong. The knot 7₁ has unknotting number 3. Its mirror image also has unknotting number 3. Their connected sum should require 6 crossing changes. It requires at most 5. The combined knot is strictly simpler than the sum of its parts.
The mechanism is topological. When two knots are connected, the junction between them creates new crossing-change possibilities that don't exist in either knot alone. A crossing change near the junction can simultaneously simplify both components. The two knots aren't independent — they share structure at their boundary, and that shared structure provides shortcuts.
The deeper surprise isn't the counterexample itself but how long it took to find. Wendt's conjecture was open for nearly 90 years. During that time, every tested pair of knots respected additivity. The failure required a specific knot (7₁) and its mirror image, chosen so that the symmetry between them creates particularly efficient cross-component simplifications at the junction. The counterexample was invisible not because it was complex — 7₁ is a standard knot in every table — but because finding the non-additive crossing-change sequence required combinatorial ingenuity that earlier approaches lacked.
The general principle: additivity is the default assumption for combined systems, and it's wrong more often than intuition suggests. When two systems share a boundary, the boundary creates interactions that can reduce combined complexity below the sum. This happens in knot theory, in team productivity (two people working together can solve problems that neither can alone, in fewer total steps), and in any system where components interact rather than merely coexist. The sum of the parts is an upper bound, not a law.