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Turaev genus and alternating decompositions

Cody W Armond and Adam M Lowrance

Algebraic & Geometric Topology 17 (2017) 793–830

We prove that the genus of the Turaev surface of a link diagram is determined by a graph whose vertices correspond to the boundary components of the maximal alternating regions of the link diagram. Furthermore, we use these graphs to classify link diagrams whose Turaev surface has genus one or two, and we prove that similar classification theorems exist for all genera.

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knot, link, Turaev genus, almost-alternating, alternating decomposition
Mathematical Subject Classification 2010
Primary: 57M25
Secondary: 57M27
Received: 9 July 2015
Revised: 15 June 2016
Accepted: 4 August 2016
Published: 14 March 2017
Cody W Armond
Department of Mathematics
Ohio State University at Mansfield
1760 University Drive
Mansfield, OH 44906
United States
Adam M Lowrance
Department of Mathematics
Vassar College
124 Raymond Ave Box 257
Poughkeepsie, NY 12604
United States