# On the Neuwirth conjecture for knots

@article{Ozawa2011OnTN, title={On the Neuwirth conjecture for knots}, author={Makoto Ozawa and J. Hyam Rubinstein}, journal={arXiv: Geometric Topology}, year={2011} }

Neuwirth asked if any non-trivial knot in the 3-sphere can be embedded in a closed surface so that the complement of the surface is a connected essential surface for the knot complement. In this paper, we examine some variations on this question and prove it for all knots up to 11 crossings except for two examples. We also establish the conjecture for all Montesinos knots and for all generalized arborescently alternating knots. For knot exteriors containing closed incompressible surfaces…

## 8 Citations

### The Neuwirth Conjecture for a family of satellite knots

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### A knot without a nonorientable essential spanning surface

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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…

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The slope conjecture relates the degree of the colored Jones polynomial of a knot to boundary slopes of essential surfaces. We develop a general approach that matches a state-sum formula for the…

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In this paper, we investigate three geometrical invariants of knots, the height, the trunk and the representativity.
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### Jones slopes and coarse volume of near-alternating links

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We study near-alternating links whose diagrams satisfy conditions generalized from the notion of semi-adequate links. We extend many of the results known for adequate knots relating their colored…

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This article is an English translation of Japanese article "Musubime to Kyokumen", Math. Soc. Japan, Sugaku Vol. 67, No. 4 (2015) 403--423. It surveys a specific area in Knot Theory concerning…

### Geometry of alternating links on surfaces

- MathematicsTransactions of the American Mathematical Society
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We consider links that are alternating on surfaces embedded in a compact 3-manifold. We show that under mild restrictions, the complement of the link decomposes into simpler pieces, generalising the…

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