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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?.

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

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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.

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  • $\begingroup$ 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. $\endgroup$
    – user113715
    Feb 11, 2015 at 6:37
  • $\begingroup$ 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). $\endgroup$ Feb 11, 2015 at 20:46
  • $\begingroup$ 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. $\endgroup$ Feb 11, 2015 at 20:51

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