Supposant qu’un nœud hyperbolique dans admet une chirurgie finie, Boyer et Zhang ont prouvé que la pente de la chirurgie doit être soit un entier, soit un demi-entier, et ils ont conjecturé que le dernier cas ne se produit pas. En utilisant les termes de correction dans l’homologie de Heegaard Floer, nous prouvons que si un noeud hyperbolique dans admet une chirurgie finie demi-entier, alors il doit avoir la même homologie de Floer des nœuds l’un des huit nœuds non-hyperboliques qui sont connus pour avoir ces chirurgies, et la variété résultante doit être l’une des dix formes de l’espace sphérique. Comme l’homologie de Floer des nœuds porte beaucoup d’informations sur le nœud, cela apporte une forte évidence à la conjecture de Boyer–Zhang.
Supposing that a hyperbolic knot in admits a finite surgery, Boyer and Zhang proved that the surgery slope must be either integral or half-integral, and they conjectured that the latter case does not happen. Using the correction terms in Heegaard Floer homology, we prove that if a hyperbolic knot in admits a half-integral finite surgery, then the knot must have the same knot Floer homology as one of the eight non-hyperbolic knots which are known to admit such surgeries, and the resulting manifold must be one of ten spherical space forms. As knot Floer homology carries a lot of information about the knot, this gives a strong evidence to Boyer–Zhang’s conjecture.
@article{AFST_2015_6_24_5_1157_0,
author = {Li, Eileen and Ni, Yi},
title = {Half-integral finite surgeries on knots in $S^3$},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {1157--1178},
publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 24},
number = {5},
year = {2015},
doi = {10.5802/afst.1479},
language = {en},
url = {http://www.numdam.org/articles/10.5802/afst.1479/}
}
TY - JOUR AU - Li, Eileen AU - Ni, Yi TI - Half-integral finite surgeries on knots in $S^3$ JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2015 SP - 1157 EP - 1178 VL - 24 IS - 5 PB - Université Paul Sabatier, Institut de mathématiques PP - Toulouse UR - http://www.numdam.org/articles/10.5802/afst.1479/ DO - 10.5802/afst.1479 LA - en ID - AFST_2015_6_24_5_1157_0 ER -
%0 Journal Article %A Li, Eileen %A Ni, Yi %T Half-integral finite surgeries on knots in $S^3$ %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2015 %P 1157-1178 %V 24 %N 5 %I Université Paul Sabatier, Institut de mathématiques %C Toulouse %U http://www.numdam.org/articles/10.5802/afst.1479/ %R 10.5802/afst.1479 %G en %F AFST_2015_6_24_5_1157_0
Li, Eileen; Ni, Yi. Half-integral finite surgeries on knots in $S^3$. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 5, pp. 1157-1178. doi : 10.5802/afst.1479. http://www.numdam.org/articles/10.5802/afst.1479/
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