Algebraic & Geometric Topology

Essential twisted surfaces in alternating link complements

Marc Lackenby and Jessica Purcell

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841258

Digital Object Identifier
doi:10.2140/agt.2016.16.3209

Mathematical Reviews number (MathSciNet)
MR3584257

Zentralblatt MATH identifier
1355.57005

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

Keywords
alternating links essential surfaces

Citation

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. https://projecteuclid.org/euclid.agt/1510841258


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