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The graded count of quasi-trees is not a knot invariant


Authors: Cody Armond and Moshe Cohen
Journal: Proc. Amer. Math. Soc. 144 (2016), 2285-2290
MSC (2010): Primary 57M25, 57M27, 57M15, 05C31, 05C10
DOI: https://doi.org/10.1090/proc/12842
Published electronically: September 24, 2015
MathSciNet review: 3460186
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Abstract: In ``A survey on the Turaev genus of knots'', Champanerkar and Kofman propose several open questions. The first asks whether the polynomial whose coefficients count the number of quasi-trees of the all-A ribbon graph obtained from a diagram with minimal Turaev genus is an invariant of the knot. We answer negatively by showing a counterexample obtained from the two diagrams of $ 8_{21}$ on the Knot Atlas and KnotScape.


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Additional Information

Cody Armond
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
Address at time of publication: Department of Math and Statistics, University of South Alabama, Mobile, Alabama 36688
Email: codyarmond@southalabama.edu

Moshe Cohen
Affiliation: Department of Electrical Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel
Email: mcohen@tx.technion.ac.il

DOI: https://doi.org/10.1090/proc/12842
Keywords: Dessin d'enfant, combinatorial map, graph on surface, spanning tree, Turaev surface
Received by editor(s): October 27, 2014
Received by editor(s) in revised form: May 21, 2015
Published electronically: September 24, 2015
Communicated by: Ken Ono
Article copyright: © Copyright 2015 American Mathematical Society