The conjecture cr(C_m\times C_n)=(m-2)n is true for all but finitely many n, for each m

Abstract

It has been long congectured that the crossing number of $C_m\times C_n$ is $(m-2)n$ for $2<m<=n$. In this paper we proved that conjecture is true for all but finitely many $n$ for each $m$. More specifically we proved conjecture for $n>=(m/2)((m+3)^2/2+1)$.The proof is largely based on the theory of arrangements introduced by Adamsson and further developed by Adamsson and Richter.