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 be a prime alternating achiral knot. Then there exists a minimal projection of in and an involution such that:
1) reverses the orientation of ;
2) ;
3) ;
4) has two fixed points on and hence reverses the orientation of .
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].
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 un noeud premier, alterné et achiral. Alors il existe une projection de dans et une involution telles que :
1) renverse l’orientation de ;
2) ;
3) ;
4) a deux points fixes sur et donc renverse l’orientation de .
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].
Nicola Ermotti 1; Cam Van Quach Hongler 1; Claude Weber 1
@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, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 21},
number = {1},
year = {2012},
doi = {10.5802/afst.1328},
mrnumber = {2954104},
zbl = {1239.57011},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1328/}
}
TY - JOUR AU - Nicola Ermotti AU - Cam Van Quach Hongler AU - Claude Weber TI - A proof of Tait’s Conjecture on prime alternating $-$achiral knots JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2012 SP - 25 EP - 55 VL - 21 IS - 1 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1328/ DO - 10.5802/afst.1328 LA - en ID - AFST_2012_6_21_1_25_0 ER -
%0 Journal Article %A Nicola Ermotti %A Cam Van Quach Hongler %A Claude Weber %T A proof of Tait’s Conjecture on prime alternating $-$achiral knots %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2012 %P 25-55 %V 21 %N 1 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1328/ %R 10.5802/afst.1328 %G en %F AFST_2012_6_21_1_25_0
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, Serie 6, Volume 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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