We prove that if is a nontrivial alternating link embedded (without crossings) in a closed surface , then has a compressing disk whose boundary intersects in no more than two points. Moreover, whenever the surface is incompressible and –incompressible in the link exterior, it can be isotoped to have a standard tube at some crossing of any reduced alternating diagram.
"Alternating links have representativity $2$." Algebr. Geom. Topol. 18 (6) 3339 - 3362, 2018. https://doi.org/10.2140/agt.2018.18.3339