DEHN SURGERY AND A-POLYNOMIAL FOR KNOTS

Title & Authors
DEHN SURGERY AND A-POLYNOMIAL FOR KNOTS
Kim, Jin-Hong;

Abstract
The Property P Conjecture States that the 3-manifold $\small{Y_r}$ obtained by Dehn surgery on a non-trivial knot in $\small{S^3}$ with surgery coefficient $\small{{\gamma}{\in}Q}$ has the non-trivial fundamental group (so not simply connected). Recently Kronheimer and Mrowka provided a proof of the Property P conjecture for the case $\small{{\gamma}={\pm}2}$ that was the only remaining case to be established for the conjecture. In particular, their results show that the two phenomena of having a cyclic fundamental group and having a homomorphism with non-cyclic image in SU(2) are quite different for 3-manifolds obtained by Dehn filings. In this paper we extend their results to some other Dehn surgeries via the A-polynomial, and provide more evidence of the ubiquity of the above mentioned phenomena.
Keywords
Dehn surgery;property P conjecture;A-polynomials;
Language
English
Cited by
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