Alternating links have representativity $2$

Abstract

We prove that if $L$ is a nontrivial alternating link embedded (without crossings) in a closed surface $F\subset S^3$, then $F$ has a compressing disk whose boundary intersects $L$ in no more than two points. Moreover, whenever the surface is incompressible and $\partial$–incompressible in the link exterior, it can be isotoped to have a standard tube at some crossing of any reduced alternating diagram.