A lower bound of the crossing number of composite knots

Abstract

Let $c(K)$ denote the crossing number of a knot $K$, and let $K_1\#K_2$ denote the connected sum of two oriented knots $K_1$ and $K_2$. As a famous old question in knot theory, whether $c(K_1\#K_2)=c(K_1)+c(K_2)$ is still unsolved. In this paper, we show that for any nontrivial knot $K=K_1\#\cdots \#K_n$, where $K_1,\dots ,K_n$ are oriented knots, the following inequality holds \[c(K)>\frac{1}{16}(c(K_1)+\cdots+c(K_n)).\] The result improves a well-known lower bound of $c(K)$ given by Lackenby.

64 pages, 55 figures