Geodesic
Knots on a positive template have a bounded number of prime factors
Sullivan, Michael C1
1Department of Mathematics (4408), Southern Illinois University, Carbondale IL 62901, USA
Algebraic and Geometric Topology, Tome 5 (2005) no. 2, pp. 563-576
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

Templates are branched 2–manifolds with semi-flows used to model “chaotic” hyperbolic invariant sets of flows on 3–manifolds. Knotted orbits on a template correspond to those in the original flow. Birman and Williams conjectured that for any given template the number of prime factors of the knots realized would be bounded. We prove a special case when the template is positive; the general case is now known to be false.

DOI : 10.2140/agt.2005.5.563
Keywords: hyperbolic flows, templates, prime knots, composite knots, positive braids

Sullivan, Michael C  1

1 Department of Mathematics (4408), Southern Illinois University, Carbondale IL 62901, USA 
@article{10_2140_agt_2005_5_563,
     author = {Sullivan, Michael C},
     title = {Knots on a positive template have a bounded number of prime factors},
     journal = {Algebraic and Geometric Topology},
     pages = {563--576},
     year = {2005},
     volume = {5},
     number = {2},
     doi = {10.2140/agt.2005.5.563},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2005.5.563/}
}
TY  - JOUR
AU  - Sullivan, Michael C
TI  - Knots on a positive template have a bounded number of prime factors
JO  - Algebraic and Geometric Topology
PY  - 2005
SP  - 563
EP  - 576
VL  - 5
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2005.5.563/
DO  - 10.2140/agt.2005.5.563
ID  - 10_2140_agt_2005_5_563
ER  - 
%0 Journal Article
%A Sullivan, Michael C
%T Knots on a positive template have a bounded number of prime factors
%J Algebraic and Geometric Topology
%D 2005
%P 563-576
%V 5
%N 2
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2005.5.563/
%R 10.2140/agt.2005.5.563
%F 10_2140_agt_2005_5_563
Sullivan, Michael C. Knots on a positive template have a bounded number of prime factors. Algebraic and Geometric Topology, Tome 5 (2005) no. 2, pp. 563-576. doi: 10.2140/agt.2005.5.563

[1] J S Birman, R F Williams, Knotted periodic orbits in dynamical system II: Knot holders for fibered knots, from: "Low-dimensional topology (San Francisco, Calif., 1981)", Contemp. Math. 20, Amer. Math. Soc. (1983) 1

[2] G Burde, H Zieschang, Knots, de Gruyter Studies in Mathematics 5, Walter de Gruyter Co. (2003)

[3] P R Cromwell, Positive braids are visually prime, Proc. London Math. Soc. $(3)$ 67 (1993) 384

[4] J Franks, R F Williams, Entropy and knots, Trans. Amer. Math. Soc. 291 (1985) 241

[5] R W Ghrist, Branched two-manifolds supporting all links, Topology 36 (1997) 423

[6] R W Ghrist, P J Holmes, M C Sullivan, Knots and links in three-dimensional flows, Lecture Notes in Mathematics 1654, Springer (1997)

[7] M Ozawa, Closed incompressible surfaces in the complements of positive knots, Comment. Math. Helv. 77 (2002) 235

[8] M C Sullivan, Composite knots in the figure-8 knot complement can have any number of prime factors, Topology Appl. 55 (1994) 261

[9] M C Sullivan, The prime decomposition of knotted periodic orbits in dynamical systems, J. Knot Theory Ramifications 3 (1994) 83

[10] M C Sullivan, Factoring positive braids via branched manifolds, Topology Proc. 30 (2006) 403

[11] R F Williams, Lorenz knots are prime, Ergodic Theory Dynam. Systems 4 (1984) 147

Cité par Sources :