Take a knot. Measure its unknotting number — the minimum cuts and reconnections needed to reduce it to a simple loop. Call it 3. Now take its mirror image — same shape, opposite handedness. Its unknotting number is also 3. Connect the two by cutting each and joining the loose ends. The result should require 6 moves to unknot: 3 for each component. This has been assumed since Wendt conjectured it in 1937. The unknotting number should be additive under composition.
Brittenham and Hermiller (University of Nebraska-Lincoln, 2026) found a counterexample. A specific knot connected to its own mirror image can be unknotted in 5 moves, possibly fewer. The combined knot is simpler than the sum of its parts.
The intuition that complexity must be additive is strong. Two problems stacked should be at least as hard as either one alone. Adding obstacles should not create shortcuts. But the mirror image isn't an arbitrary obstacle — it's precisely organized to interact with the original's structure. The two knots share every crossing pattern, reflected. During unknotting, a move that resolves a crossing in the original can simultaneously resolve a corresponding crossing in the mirror, because the mirror placed that crossing in the exact position where it becomes reachable. The reflection creates access that random addition wouldn't.
This distinguishes the result from simple cancellation. The knot and its mirror don't cancel to nothing — the composite is still complex, still requires at least 5 moves. The mirror doesn't neutralize the original. It provides structural correspondence that the unknotting process can exploit. The complication is real. The simplification comes from the relationship between the complication and what was already there.
The operative word is additive. Unknotting number is a minimum — the fewest moves needed. Minimization problems are the ones most vulnerable to non-additivity, because finding the optimum depends on the full landscape of possible moves, and combining structures can reshape that landscape. Adding a mirror image adds crossings but also adds new sequences of moves that weren't available in either knot alone. Some of those new sequences are shorter than the sum of the individual minima.
The broader principle: when a complexity measure involves optimization over a space of strategies, combining structures can expand the strategy space in ways that reduce the optimum. The components interact not by canceling difficulty but by creating paths through it that neither component alone could offer. The complication carries its own shortcut — but only if it's the right complication. An arbitrary knot composed with the original gives no such guarantee. The mirror's precision is what makes the simplification possible.
As mathematician Kristen Hendricks noted, the result says “our notions of complexity could have problems.” The deeper issue is that unknotting number was treated as a property of the knot when it's actually a property of the knot's relationship to the full space of moves. Adding another knot changes that space. The measure looked intrinsic. It was relational.