Dehn surgery, the fundamental group and SU(2)

By P. Kronheimer, T. Mrowka. 2003

83 citations recorded by the source corpus.

Let K be a non-trivial knot in the 3-sphere and let Y(r) be the 3-manifold obtained by surgery on K with surgery-coefficient a rational number r. We show that there is a homomorphism from the fundamental group of Y(r) to SU(2) with non-cyclic image if r is less than or equal to 2.

Subfields: Genetics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2546b98bf37392416b9741a30bb82f440f0700cd

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Citations in the corpus

Knot contact homology (2011, methodology): “For example, the proof in [25] that the framed knot DGA detects the unknot uses work of Dunfield and Garoufalidis [7], building on some deep gauge-theoretic results of Kronheimer and Mrowka [20].”
A topological introduction to knot contact homology (2012, unknown): “However, the proof of Corollary 4.10 uses only the Loop Theorem, rather than the deep Kronheimer–Mrowka result that leads to Corollary 5.10.”
Mixed 3-manifolds are virtually special (2012, background): “has a faithful representation in SL(n;Z) for some n. Note that existence of a non-abelian representation of any nontrivial knot complement group into SU(2) is a well-known result of Kronheimer–Mrowka [KM04]. MIXED 3–MANIFOLDS ARE VIRTUALLY SPECIAL 3 Organization. As explained in Section 2, the proof”
Knot homology groups from instantons (2008, result): “f critical points is empty. For the unknot therefore, the group LIN ∗ (K) is zero. The same applies to an “unknot” in any Y, i.e. a knot that bounds a disk. In part because of the results of [24] and [23], it is natural to conjecture that LI∗(Y,K) is zero only if K is an unknot. An examination of th”
Knots, sutures, and excision (2008, result): “This proposition provides an alternative proof for results from [17] and [16].”
The pillowcase and perturbations of traceless representations of knot groups (2013, background): “ial torus knot is always non-zero (see. e.g. [16]) from which it follows that R(S3;K) always contains at least one point on a non-abelian arc. A much deeper result is the theorem of Kronheimer-Mrowka [20] that every non-trivial knot in S3 admits a non-abelian traceless representation. Applying Propo”