# Seifert Klein bottles for knots with common boundary slopes

@article{ValdezSnchez2004SeifertKB, title={Seifert Klein bottles for knots with common boundary slopes}, author={Luis G. Valdez-S{\'a}nchez}, journal={arXiv: Geometric Topology}, year={2004} }

We consider the question of how many essential Seifert Klein bottles with common boundary slope a knot in S^3 can bound, up to ambient isotopy. We prove that any hyperbolic knot in S^3 bounds at most six Seifert Klein bottles with a given boundary slope. The Seifert Klein bottles in a minimal projection of hyperbolic pretzel knots of length 3 are shown to be unique and pi_1-injective, with surgery along their boundary slope producing irreducible toroidal manifolds. The cable knots which bound…

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## References

SHOWING 1-10 OF 35 REFERENCES

### Complements of Minimal Spanning Surfaces of Knots are Not Unique

- Mathematics
- 1970

Every polyhedral simple closed curve in S3 bounds an orientable surface [2]; that is, if k is a knot, then there exists an orientable surface S such that D(S) = k. The minimal genus of all such…

### Dehn surgeries on knots creating essential tori, II

- Mathematics
- 2000

In this paper, which is a sequel to [GLul], we continue our study of when Dehn surgery on a hyperbolic knot K in S can yield a manifold that contains an incompressible torus. Let E(K) denote the…

### Dehn surgery on knots

- Medicine
- 1985

In [D], Dehn considered the following method for constructing 3-manifolds: remove a solid torus neighborhood N(K) of some knot X in the 3-sphere S and sew it back differently, showing that one could obtain infinitely many non-simply-connected homology spheres o in this way.

### Incompressible spanning surfaces and maximal fibred submanifolds for a knot

- Mathematics
- 1992

In a previous paper [7] we introduced and studied maximal fibred submanifolds of the exterior of a knot. In this paper we shall investigate the relation of these submanifolds to the number of…

### Crosscap number two knots in $S^3$ with (1,1) decompositions

- Mathematics
- 2005

M. Scharlemann has recently proved that any genus one tunnel number one knot is either a satellite or 2-bridge knot, as conjectured by H. Goda and M. Teragaito; all such knots admit a (1,1)…

### Knots with infinitely many minimal spanning surfaces

- Mathematics
- 1977

Introduction. If K is a polygonal representative of a tame knot in S3, then K is spahned by a polyhedral, orientable surface [10, ?7], [23]; an orientable spanning surface of smallest possible genus…

### Foliations and the topology of 3-manifolds

- Mathematics
- 1983

In this announcement we discuss the close relationship between the topology of 3-manifolds and the foliations that is possesses. We will introduce and state the main result, then use it and the ideas…