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Additivity of the crossing number under connected sum

Establish whether the crossing number is additive under connected sum; prove that for any knots K and K' in S^3, the equality c(K # K') = c(K) + c(K') holds.

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Background

The crossing number c(K) is the minimal number of crossings among all diagrams of K. While widely believed to be additive under connected sum, no proof is known. The authors note the parallel to the unknotting number additivity question and emphasize the lack of resolution.

A resolution would impact knot tabulation and complexity analyses, clarifying how crossing numbers behave under composition and informing algorithmic and combinatorial approaches in knot theory.

References

Note that it is also open whether the crossing number is additive under connected sum, though this is widely believed to be true.

— The unknotting number, hard unknot diagrams, and reinforcement learning (2409.09032 - Applebaum et al., 13 Sep 2024) in Subsection “Additivity of unknotting number,” Section 1 (Introduction)