additivity of crossing number of composite knots Let $K_1 \# K_2$ denotes the connected sum of tow knots $K_1$ and $K_2$. The crossing number is the minimal number of crossings among all knot diagrams,  denoted by $c(K)$. It is conjectured that $c(K_1 \# K_2)=c(K_1) + c(K_2)$. Is this conjecture settled down or still open?   Is there any counterexample?. 
 A: It's an open problem in general (there are no counterexamples, to my knowledge), but consider these partial results:


*

*$c(K_1 \# K_2) \le c(K_1) + c(K_2)$ for any two knots $K_1$, $K_2$.  To see this, suppose that you have diagrams for $K_1$ and $K_2$ with minimal number of crossings, say $n_1$ and $n_2$, respectively.  Their connect sum is a diagram for $K_1 \# K_2$ with $n_1 + n_2$ crossings, although this may not be minimal.

*Assuming that the knots are alternating, this is a theorem of Kauffman, Murasugi, and Thistlethwaite using knot polynomials.  (See MR:0999974.)

*There's a larger class of adequate knots that includes alternating knots for which the conjecture holds.

*The conjecture holds for torus links.  (See arXiv:0303273)

*There's a bound in the other direction of the form $c(K_1 \# K_2) \ge \frac{1}{N} \bigl( c(K_1) + c(K_2) \bigr)$ for some constant $N$.  In one result, $N = 152$, but this can presumably be improved.  Of course, $N = 1$ would prove the conjecture in full generality.  (See MR:2574742.)


This question is as old as knot theory and Tait's conjectures.
