Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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DEHN SURGERY AND A-POLYNOMIAL FOR KNOTS

  • Kim, Jin-Hong (Department of Mathematics, KAIST)
  • Published : 2006.08.01

Abstract

The Property P Conjecture States that the 3-manifold $Y_r$ obtained by Dehn surgery on a non-trivial knot in $S^3$ with surgery coefficient ${\gamma}{\in}Q$ 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 ${\gamma}={\pm}2$ 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

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