Volume 18, issue 6 (2018)

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Alternating links have representativity $2$

Thomas Kindred

Algebraic & Geometric Topology 18 (2018) 3339–3362
Abstract

We prove that if $L$ is a nontrivial alternating link embedded (without crossings) in a closed surface $F\subset {S}^{3}\phantom{\rule{0.3em}{0ex}}$, 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.

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Keywords
alternating knot, alternating link, representativity, closed surface, compressing disk
Primary: 57M25
Secondary: 57M50