Algebraic & Geometric Topology

Essential twisted surfaces in alternating link complements

Marc Lackenby and Jessica Purcell

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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.

Article information

Algebr. Geom. Topol., Volume 16, Number 6 (2016), 3209-3270.

Received: 11 December 2014
Revised: 23 March 2016
Accepted: 21 April 2016
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

alternating links essential surfaces


Lackenby, Marc; Purcell, Jessica. Essential twisted surfaces in alternating link complements. Algebr. Geom. Topol. 16 (2016), no. 6, 3209--3270. doi:10.2140/agt.2016.16.3209.

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