Geodesic
Framed holonomic knots
Ekholm, Tobias1 ; Wolff, Maxime2
1Department of Mathematics, Uppsala University, PO Box 480, 751 06 Uppsala, Sweden
2Département de Mathématiques et Informatique, Ecole Normale Supérieure de Lyon, 46 allée d’Italie, 69364 Lyon Cédex 07, France
Algebraic and Geometric Topology, Tome 2 (2002) no. 1, pp. 449-463
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

A holonomic knot is a knot in 3–space which arises as the 2–jet extension of a smooth function on the circle. A holonomic knot associated to a generic function is naturally framed by the blackboard framing of the knot diagram associated to the 1–jet extension of the function. There are two classical invariants of framed knot diagrams: the Whitney index (rotation number) W and the self linking number S. For a framed holonomic knot we show that W is bounded above by the negative of the braid index of the knot, and that the sum of W and |S| is bounded by the negative of the Euler characteristic of any Seifert surface of the knot. The invariant S restricted to framed holonomic knots with W = m, is proved to split into n, where n is the largest natural number with n ≤ |m| 2 , integer invariants. Using this, the framed holonomic isotopy classification of framed holonomic knots is shown to be more refined than the regular isotopy classification of their diagrams.

DOI : 10.2140/agt.2002.2.449
Keywords: framing, holonomic knot, Legendrian knot, self-linking number, Whitney index

Ekholm, Tobias  1   ; Wolff, Maxime  2

1 Department of Mathematics, Uppsala University, PO Box 480, 751 06 Uppsala, Sweden
2 Département de Mathématiques et Informatique, Ecole Normale Supérieure de Lyon, 46 allée d’Italie, 69364 Lyon Cédex 07, France 
@article{10_2140_agt_2002_2_449,
     author = {Ekholm, Tobias and Wolff, Maxime},
     title = {Framed holonomic knots},
     journal = {Algebraic and Geometric Topology},
     pages = {449--463},
     year = {2002},
     volume = {2},
     number = {1},
     doi = {10.2140/agt.2002.2.449},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2002.2.449/}
}
TY  - JOUR
AU  - Ekholm, Tobias
AU  - Wolff, Maxime
TI  - Framed holonomic knots
JO  - Algebraic and Geometric Topology
PY  - 2002
SP  - 449
EP  - 463
VL  - 2
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2002.2.449/
DO  - 10.2140/agt.2002.2.449
ID  - 10_2140_agt_2002_2_449
ER  - 
%0 Journal Article
%A Ekholm, Tobias
%A Wolff, Maxime
%T Framed holonomic knots
%J Algebraic and Geometric Topology
%D 2002
%P 449-463
%V 2
%N 1
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2002.2.449/
%R 10.2140/agt.2002.2.449
%F 10_2140_agt_2002_2_449
Ekholm, Tobias; Wolff, Maxime. Framed holonomic knots. Algebraic and Geometric Topology, Tome 2 (2002) no. 1, pp. 449-463. doi: 10.2140/agt.2002.2.449

[1] D Bennequin, Entrelacements et équations de Pfaff, from: "Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982)", Astérisque, Soc. Math. France (1983) 87

[2] J S Birman, N C Wrinkle, Holonomic and Legendrian parametrizations of knots, J. Knot Theory Ramifications 9 (2000) 293

[3] L H Kauffman, Knots and physics, Series on Knots and Everything, World Scientific Publishing Co. Inc. (1991)

[4] B Trace, On the Reidemeister moves of a classical knot, Proc. Amer. Math. Soc. 89 (1983) 722

[5] V A Vassiliev, Holonomic links and Smale principles for multisingularities, J. Knot Theory Ramifications 6 (1997) 115

[6] H Whitney, On regular closed curves in the plane, Compositio Math. 4 (1937) 276

Cité par Sources :