Mathematics Colloquia and Seminars

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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, many questions of much weaker conditions are still open. For instance, it is not known whether $Cr(K_1\# K_2)\ge Cr(K_1)$ or $Cr(K_1\# K_2)\ge Cr(K_2)$ holds in general, here $K_1\# K_2$ is the connected sum of $K_1$ and $K_2$ and $Cr(K)$ stands for the crossing number of the link $K$. However, for alternating links $K_1$ and $K_2$, $Cr(K_1\# K_2)=Cr(K_1)+Cr(K_2)$ does hold. In this talk, I will show that there exist a wide class of links over which the crossing number is additive under the connected sum operation. I will then show that the torus knot family is within this class of knots together with many alternating knots. Consequently, $$ Cr(T_1\# T_2\# \cdots \#T_m)=Cr(T_1)+Cr(T_2)+\cdots +Cr(T_m) $$ as long as each $T_j$ is a torus knot or an alternating knot in this class. Furthermore, if $K_1$ is a connected sum of any given number of torus knots and $K_2$ is a non-trivial knot, we prove that $Cr(K_1\# K_2)\ge Cr(K_1)+3$.