The crossing-number additivity conjecture for knots

Let $\tau_1$ and $\tau_2$ be knots, let \tau_1#\tau_2 denote their connected sum, and let $\operatorname{cr}(K)$ denote the crossing number of a knot $K$. Crossing-number additivity conjecture. The crossing number of knots is additive with respect to connected sum:

\operatorname{cr}(\tau_1#\tau_2)=\operatorname{cr}(\tau_1)+\operatorname{cr}(\tau_2).

This is described as a longstanding conjecture and is known for alternating and adequate knots, but remains open in general; the paper shows that it contradicts the generic hyperbolicity conjecture.