Additivity of the unknotting number under connected sum

Establish additivity of the unknotting number under connected sum for all knots in S^3; prove that for any knots K and K', the equality u(K # K') = u(K) + u(K') holds.

Background

The unknotting number u(K) is the minimal number of crossing changes needed to turn a knot K into the unknot. Whether u is additive under connected sum has long been considered fundamental yet elusive. The paper describes limited theoretical evidence (e.g., lower bounds such as Scharlemann's u(K # K') ≥ 2 for nontrivial K and K') and discusses a reinforcement-learning-driven search for counterexamples, which was unsuccessful, leaving the question open.

Resolving additivity would have broad implications for computing unknotting numbers, since it would enable determination of intermediate unknotting numbers along minimal sequences for connected sums. The authors highlight that even for naturally tractable classes, such as torus knots with opposite-sign signatures, additivity remains unknown.

References

An old open question is whether the unknotting number is additive under connected sum.

— The unknotting number, hard unknot diagrams, and reinforcement learning  (2409.09032 - Applebaum et al., 2024) in Subsection “Additivity of unknotting number,” Section 1 (Introduction)

Whether there exists a knot which does not possess a symbiont is unknown, and provides a deeply interesting open problem.

— Obstructing Unknotting Number Two For Some Alternating Knots  (2608.25256 - Gebel et al., 26 Aug 2026) in Introduction