Geodesic
Unexpected local minima in the width complexes for knots
Zupan, Alexander1
1Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City IA 52242, USA
Algebraic and Geometric Topology, Tome 11 (2011) no. 2, pp. 1097-1105
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

In [Pacific J. Math. 239 (2009) 135–156], Schultens defines the width complex for a knot in order to understand the different positions a knot can occupy in S3 and the isotopies between these positions. She poses several questions about these width complexes; in particular, she asks whether the width complex for a knot can have local minima that are not global minima. In this paper, we find an embedding of the unknot 01 that is a local minimum but not a global minimum in the width complex for 01, resolving a question of Scharlemann. We use this embedding to exhibit for any knot K infinitely many distinct local minima that are not global minima of the width complex for K.

DOI : 10.2140/agt.2011.11.1097
Classification : 57M25, 57M27
Keywords: width complex, thin position, unknot

Zupan, Alexander  1

1 Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City IA 52242, USA 
@article{10_2140_agt_2011_11_1097,
     author = {Zupan, Alexander},
     title = {Unexpected local minima in the width complexes for knots},
     journal = {Algebraic and Geometric Topology},
     pages = {1097--1105},
     year = {2011},
     volume = {11},
     number = {2},
     doi = {10.2140/agt.2011.11.1097},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2011.11.1097/}
}
TY  - JOUR
AU  - Zupan, Alexander
TI  - Unexpected local minima in the width complexes for knots
JO  - Algebraic and Geometric Topology
PY  - 2011
SP  - 1097
EP  - 1105
VL  - 11
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2011.11.1097/
DO  - 10.2140/agt.2011.11.1097
ID  - 10_2140_agt_2011_11_1097
ER  - 
%0 Journal Article
%A Zupan, Alexander
%T Unexpected local minima in the width complexes for knots
%J Algebraic and Geometric Topology
%D 2011
%P 1097-1105
%V 11
%N 2
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2011.11.1097/
%R 10.2140/agt.2011.11.1097
%F 10_2140_agt_2011_11_1097
Zupan, Alexander. Unexpected local minima in the width complexes for knots. Algebraic and Geometric Topology, Tome 11 (2011) no. 2, pp. 1097-1105. doi: 10.2140/agt.2011.11.1097

[1] F Bonahon, J P Otal, Scindements de Heegaard des espaces lenticulaires, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982) 585

[2] D Gabai, Foliations and the topology of 3–manifolds. III, J. Differential Geom. 26 (1987) 479

[3] J P Otal, Présentations en ponts du nœud trivial, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982) 553

[4] M Ozawa, Nonminimal bridge positions of torus knots are stabilized

[5] M Scharlemann, Thin position in the theory of classical knots, from: "Handbook of knot theory" (editors W Menasco, M Thistlethwaite), Elsevier (2005) 429

[6] M Scharlemann, A Thompson, Thin position for 3–manifolds, from: "Geometric topology (Haifa, 1992)" (editors C Gordon, Y Moriah, B Wajnryb), Contemp. Math. 164, Amer. Math. Soc. (1994) 231

[7] H Schubert, Über eine numerische Knoteninvariante, Math. Z. 61 (1954) 245

[8] J Schultens, Additivity of bridge numbers of knots, Math. Proc. Cambridge Philos. Soc. 135 (2003) 539

[9] J Schultens, Width complexes for knots and 3–manifolds, Pacific J. Math. 239 (2009) 135

[10] A Thompson, Thin position and bridge number for knots in the 3–sphere, Topology 36 (1997) 505

[11] F Waldhausen, Heegaard-Zerlegungen der 3–Sphäre, Topology 7 (1968) 195

[12] A Zupan, Properties of knots preserved by cabling

Cité par Sources :