Geodesic
Flipping bridge surfaces and bounds on the stable bridge number
Johnson, Jesse1 ; Tomova, Maggy2
1Mathematics Department, Oklahoma State University, 401 Mathematical Sciences, Stillwater OK 74078-1058, USA
2Department of Mathematics, The University of Iowa, Iowa City IA 52242, USA
Algebraic and Geometric Topology, Tome 11 (2011) no. 4, pp. 1987-2005
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

We show that if K is a knot in S3 and Σ is a bridge sphere for K with high distance and 2n punctures, the number of perturbations of K required to interchange the two balls bounded by Σ via an isotopy is n. We also construct a knot with two different bridge spheres with 2n and 2n − 1 bridges respectively for which any common perturbation has at least 3n − 4 bridges. We generalize both of these results to bridge surfaces for knots in any 3–manifold.

DOI : 10.2140/agt.2011.11.1987
Keywords: stable Euler characteristic, flipping genus, bridge surface, common stabilization, knot distance, bridge position, Heegaard splitting, strongly irreducible, weakly incompressible

Johnson, Jesse  1   ; Tomova, Maggy  2

1 Mathematics Department, Oklahoma State University, 401 Mathematical Sciences, Stillwater OK 74078-1058, USA
2 Department of Mathematics, The University of Iowa, Iowa City IA 52242, USA 
@article{10_2140_agt_2011_11_1987,
     author = {Johnson, Jesse and Tomova, Maggy},
     title = {Flipping bridge surfaces and bounds on the stable bridge number},
     journal = {Algebraic and Geometric Topology},
     pages = {1987--2005},
     year = {2011},
     volume = {11},
     number = {4},
     doi = {10.2140/agt.2011.11.1987},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2011.11.1987/}
}
TY  - JOUR
AU  - Johnson, Jesse
AU  - Tomova, Maggy
TI  - Flipping bridge surfaces and bounds on the stable bridge number
JO  - Algebraic and Geometric Topology
PY  - 2011
SP  - 1987
EP  - 2005
VL  - 11
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2011.11.1987/
DO  - 10.2140/agt.2011.11.1987
ID  - 10_2140_agt_2011_11_1987
ER  - 
%0 Journal Article
%A Johnson, Jesse
%A Tomova, Maggy
%T Flipping bridge surfaces and bounds on the stable bridge number
%J Algebraic and Geometric Topology
%D 2011
%P 1987-2005
%V 11
%N 4
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2011.11.1987/
%R 10.2140/agt.2011.11.1987
%F 10_2140_agt_2011_11_1987
Johnson, Jesse; Tomova, Maggy. Flipping bridge surfaces and bounds on the stable bridge number. Algebraic and Geometric Topology, Tome 11 (2011) no. 4, pp. 1987-2005. doi: 10.2140/agt.2011.11.1987

[1] D Bachman, Heegaard splittings of sufficiently complicated $3$–manifolds I: Stabilization

[2] T Evans, High distance Heegaard splittings of $3$–manifolds, Topology Appl. 153 (2006) 2631

[3] J Hass, A Thompson, W Thurston, Stabilization of Heegaard splittings, Geom. Topol. 13 (2009) 2029

[4] C Hayashi, K Shimokawa, Heegaard splittings of trivial arcs in compression bodies, J. Knot Theory Ramifications 10 (2001) 71

[5] J Johnson, Bounding the stable genera of Heegaard splittings from below, J. Topol. 3 (2010) 668

[6] K Reidemeister, Zur dreidimensionalen Topologie, Abh. Math. Sem. Univ. Hamburg 11 (1933) 189

[7] M Scharlemann, M Tomova, Uniqueness of bridge surfaces for $2$–bridge knots, Math. Proc. Cambridge Philos. Soc. 144 (2008) 639

[8] J Singer, Three-dimensional manifolds and their Heegaard diagrams, Trans. Amer. Math. Soc. 35 (1933) 88

[9] K Takao, A refinement of Johnson's bounding for the stable genera of Heegaard splittings, Osaka J. Math. 48 (2011) 251

[10] M Tomova, Multiple bridge surfaces restrict knot distance, Algebr. Geom. Topol. 7 (2007) 957

[11] M Tomova, Thin position for knots in a $3$–manifold, J. Lond. Math. Soc. $(2)$ 80 (2009) 85

Cité par Sources :