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Incompressibility criteria for spun-normal surfaces

Authors: Nathan M. Dunfield and Stavros Garoufalidis
Journal: Trans. Amer. Math. Soc. 364 (2012), 6109-6137
MSC (2010): Primary 57N10; Secondary 57M25, 57M27
Published electronically: May 18, 2012
MathSciNet review: 2946944
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Abstract: We give a simple sufficient condition for a spun-normal surface in an ideal triangulation to be incompressible, namely that it is a vertex surface with nonempty boundary which has a quadrilateral in each tetrahedron. While this condition is far from being necessary, it is powerful enough to give two new results: the existence of alternating knots with noninteger boundary slopes, and a proof of the Slope Conjecture for a large class of 2-fusion knots.

While the condition and conclusion are purely topological, the proof uses the Culler-Shalen theory of essential surfaces arising from ideal points of the character variety, as reinterpreted by Thurston and Yoshida. The criterion itself comes from the work of Kabaya, which we place into the language of normal surface theory. This allows the criterion to be easily applied, and gives the framework for proving that the surface is incompressible.

We also explore which spun-normal surfaces arise from ideal points of the deformation variety. In particular, we give an example where no vertex or fundamental surface arises in this way.

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Additional Information

Nathan M. Dunfield
Affiliation: Department of Mathematics, MC-382, University of Illinois, 1409 W. Green Street, Urbana, Illinois 61801

Stavros Garoufalidis
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160

Keywords: Boundary slopes, normal surface, character variety, Jones slopes, 2-fusion knot, alternating knots
Received by editor(s): February 23, 2011
Received by editor(s) in revised form: March 2, 2011, and June 10, 2011
Published electronically: May 18, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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