# Alternating links have representativity 2

## TL;DR

This paper proves that non-trivial alternating links embedded in closed surfaces have compressing disks intersecting the link in at most two points, and describes isotoping incompressible surfaces to standard positions at crossings.

## Contribution

It establishes a bound on compressing disks for alternating links in surfaces and characterizes their isotopy to standard forms at crossings.

## Key findings

- Existence of compressing disks intersecting the link in at most two points.
- Incompressible and boundary-incompressible surfaces can be isotoped to standard tubes.
- Results apply to non-trivial alternating links in closed surfaces.

## Abstract

We prove that if $L$ is a non-trivial 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.