Incompressibility criteria for spun-normal surfaces

Nathan M. Dunfield, Stavros Garoufalidis

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
Pages (from-to)6109-6137
Number of pages29
JournalTransactions of the American Mathematical Society
Volume364
Issue number11
DOIs
StatePublished - 2012

Keywords

  • 2-fusion knot
  • Alternating knots
  • Boundary slopes
  • Character variety
  • Jones slopes
  • Normal surface

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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