# 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}
}```
• Published 14 March 2011
• Mathematics
• arXiv: Geometric Topology
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…
• Mathematics
Journal of Knot Theory and Its Ramifications
• 2019
Let [Formula: see text] be a nontrivial knot in [Formula: see text]. It was conjectured that there exists a Neuwirth surface for [Formula: see text]. That is, a closed surface in [Formula: see text]
• M. Ozawa
• Mathematics
Journal of the Australian Mathematical Society
• 2015
It is a well-known procedure for constructing a torus knot or link that first we prepare an unknotted torus and meridian disks in its complementary solid tori, and second we smooth the intersections
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
• Mathematics
International Journal of Mathematics
• 2020
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
• Mathematics
Journal of the Mathematical Society of Japan
• 2019
In this paper, we investigate three geometrical invariants of knots, the height, the trunk and the representativity. First, we give a conterexample for the conjecture which states that the height is
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
• M. Ozawa
• Mathematics
Sugaku Expositions
• 2019
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
• Mathematics
Transactions of the American Mathematical Society
• 2020
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

## References

SHOWING 1-10 OF 36 REFERENCES

• Mathematics
• 2000
It is well known that for many knot classes in the 3-sphere, every closed incompressible surface in their complements contains an essential loop which is isotopic into the boundary of the knot
• Mathematics
• 2010
In this article we study a partial ordering on knots in S 3 where K1 K2 if there is an epimorphism from the knot group of K1 onto the knot group of K2 which preserves peripheral structure. If K1 is a
• Mathematics
• 1984
This generalizes and strengthens the main theorem of [13]. Note that the hypothesis of Theorem 1 is satisfied whenever M is a knot manifold, i.e. the complement of an open tubular neighborhood of a
• M. Ozawa
• Mathematics
Journal of the Australian Mathematical Society
• 2011
Abstract We study a canonical spanning surface obtained from a knot or link diagram, depending on a given Kauffman state. We give a sufficient condition for the surface to be essential. By using the
• Mathematics
• 2012
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
• Mathematics
• 1985
To each rational number p/q, with q odd, there is associated the 2-bridge knot Kp/q shown in Fig. 1. QI bl Fig. 1. The 2-bridge knot Kp/q In (a), the central grid consists of lines of slope +p/q,