Illinois Journal of Mathematics

A knot without a nonorientable essential spanning surface

Nathan M. Dunfield

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This note gives the first example of a hyperbolic knot in the 3-sphere that lacks a nonorientable essential spanning surface; this disproves the Strong Neuwirth Conjecture formulated by Ozawa and Rubinstein. Moreover, this knot has no even strict boundary slopes, disproving the Even Boundary Slope Conjecture of the same authors. The proof is a rigorous calculation using Thurston’s spun-normal surfaces in the spirit of Haken’s original normal surface algorithms.

Article information

Illinois J. Math., Volume 60, Number 1 (2016), 179-184.

Received: 23 September 2015
Revised: 9 October 2016
First available in Project Euclid: 21 June 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M


Dunfield, Nathan M. A knot without a nonorientable essential spanning surface. Illinois J. Math. 60 (2016), no. 1, 179--184. doi:10.1215/ijm/1498032029.

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