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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Essential twisted surfaces in alternating link complements

Marc Lackenby and Jessica S Purcell

Algebraic & Geometric Topology 16 (2016) 3209–3270
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.

Keywords
alternating links, essential surfaces
Mathematical Subject Classification 2010
Primary: 57M25
References
Publication
Received: 11 December 2014
Revised: 23 March 2016
Accepted: 21 April 2016
Published: 15 December 2016
Authors
Marc Lackenby
Mathematical Institute
University of Oxford
Radcliffe Observatory Quarter
Woodstock Road
Oxford
OX2 6GG
United Kingdom
http://people.maths.ox.ac.uk/lackenby/
Jessica S Purcell
School of Mathematical Sciences
Monash University
9 Rainforest Walk
Room 401
Monash University VIC 3800
Australia
http://users.monash.edu/~jpurcell/