The Additivity of Crossing Numbers

It has long been conjectured that the crossing numbers of links are additive under the connected sum of links. This is a difficult problem in knot theory and has been open for more than 100 years. In fact, we do not even know that Cr(K1#K2) = Cr(K1) or Cr(K1#K2) = Cr(K2) holds in general, here K1#K2 is the connected sum of K1 and K2 and Cr(K) stands for the crossing number of the link K. The best known result to date is the following: if K1 and K2 are any two alternating links, then Cr(K1#K2) = Cr(K1) + Cr(K2), A less ambitious question asks for what link families the crossing number is additive. In particular, one wonders if this conjecture holds for the well known torus knot family. In this paper, we show that there exist a wide class of links over which the crossing number is additive under the connected sum operation. We then show that the torus knot family is within that class. Consequently, we show that Cr(T1#T2# · · ·#Tm) = Cr(T1) + Cr(T2) + · · ·+ Cr(Tm). for any m ≥ 2 torus knots T1, T2, ..., Tm. Furthermore, if K1 is a connected sum of any given number of torus knots and K2 is a non-trivial knot, we prove that Cr(K1#K2) ≥ Cr(K1) + 3.