additivity of crossing number of composite knots

Question

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?.

Answer 1

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.