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?.

• «It i conjectured» by whom? Can you give references? That always make a question better! Feb 8, 2015 at 11:12

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.

• Thank you Sammy for your answer. But really I wonder why the diagram of connected sum of two knots with $n_1 + n_2$ crossings is not minimal. I see that we can not apply any of Reidemeister moves to the resulting diagram. Feb 11, 2015 at 6:37
• I, too, would like to know why the connected sum of two minimal diagrams is minimal if it's true. It is an open problem (nobody knows if it's true). Feb 11, 2015 at 20:46
• Regarding your comment "I see that we can not apply any of Reidemeister moves to the resulting diagram": you might be overestimating your intuition about Reid. moves. There are some mildly counterintuitive truths lurking in the web of Reid. moves. For instance, there exist diagrams of a knot (of the same number of crossings) that can only be obtained, one from the other, through a sequence of Reid. moves that temporarily increase the number of crossings. Feb 11, 2015 at 20:51