Incompressibility criteria for spunnormal surfaces
Abstract
We give a simple sufficient condition for a spunnormal 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 2fusion knots. While the condition and conclusion are purely topological, the proof uses the CullerShalen 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 spunnormal 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2011
 arXiv:
 arXiv:1102.4588
 Bibcode:
 2011arXiv1102.4588D
 Keywords:

 Mathematics  Geometric Topology;
 57N10;
 57M25;
 57M27
 EPrint:
 37 pages, 8 figures. V2: New remark in Section 9.1, additional references