Mathematics – Geometric Topology
Scientific paper
2011-02-22
Mathematics
Geometric Topology
37 pages, 8 figures. V2: New remark in Section 9.1, additional references; V3 Minor edits, to appear in Trans. Amer. Math. Soc
Scientific paper
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 non-empty 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 non-integer 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.
Dunfield Nathan M.
Garoufalidis Stavros
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