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A proof of Tait’s Conjecture on prime alternating -achiral knots
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 21 (2012) no. 1, pp. 25-55.

Nous nous intéressons dans cet article à la chiralité des noeuds alternés. Nous appelons Conjecture de Tait sur les noeuds alternés négativement achiraux l’affirmation suivante :

Soit K un noeud premier, alterné et -achiral. Alors il existe une projection Π de K dans S 2 S 3 et une involution ϕ:S 3 S 3 telles que :

1) ϕ renverse l’orientation de S 3  ;

2) ϕ(S 2 )=S 2  ;

3) ϕ(Π)=Π ;

4) ϕ a deux points fixes sur Π et donc renverse l’orientation de K.

Le but de cet article est de démontrer cette affirmation.

Les prémices de cette conjecture se trouvent dans les articles de Peter Tait et de Mary Haseman. Pour plus de détails sur les origines des questions de chiralité des noeuds voir [16].

In this paper we are interested in symmetries of alternating knots, more precisely in those related to achirality. We call the following statement Tait’s Conjecture on alternating -achiral knots:

Let K be a prime alternating -achiral knot. Then there exists a minimal projection Π of K in S 2 S 3 and an involution ϕ:S 3 S 3 such that:

1) ϕ reverses the orientation of S 3 ;

2) ϕ(S 2 )=S 2 ;

3) ϕ(Π)=Π;

4) ϕ has two fixed points on Π and hence reverses the orientation of K.

The purpose of this paper is to prove this statement.

For the historical background of the conjecture in Peter Tait’s and Mary Haseman’s papers see [16].

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/afst.1328
@article{AFST_2012_6_21_1_25_0,
     author = {Nicola Ermotti and Cam Van Quach Hongler and Claude Weber},
     title = {A proof of {Tait{\textquoteright}s} {Conjecture} on prime alternating $-$achiral knots},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {25--55},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 21},
     number = {1},
     year = {2012},
     doi = {10.5802/afst.1328},
     zbl = {1239.57011},
     mrnumber = {2954104},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1328/}
}
Nicola Ermotti; Cam Van Quach Hongler; Claude Weber. A proof of Tait’s Conjecture on prime alternating $-$achiral knots. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 21 (2012) no. 1, pp. 25-55. doi : 10.5802/afst.1328. https://afst.centre-mersenne.org/articles/10.5802/afst.1328/

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