Essential twisted surfaces in alternating link complements

Marc Lackenby, Jessica Purcell

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2 Citations (Scopus)


Checkerboard surfaces in alternating link complements are used frequently to determine information about the link. However, when many crossings are added to a single twist region of a link diagram, the geometry of the link complement stabilizes (approaches a geometric limit), but a corresponding checkerboard surface increases in complexity with crossing number. In this paper, we generalize checkerboard surfaces to certain immersed surfaces, called twisted checkerboard surfaces, whose geometry better reflects that of the alternating link in many cases. We describe the surfaces, show that they are essential in the complement of an alternating link, and discuss their properties, including an analysis of homotopy classes of arcs on the surfaces in the link complement.

Original languageEnglish
Pages (from-to)3209-3270
Number of pages62
JournalAlgebraic and Geometric Topology
Issue number6
Publication statusPublished - 15 Dec 2016


  • alternating links
  • essential surfaces

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