Small Dehn surgery and SU(2)

Abstract

We prove that the fundamental group of 3-surgery on a nontrivial knot in the 3-sphere always admits an irreducible SU(2)-representation. This answers a question of Kronheimer and Mrowka dating from their work on the Property P conjecture. An important ingredient in our proof is a relationship between instanton Floer homology and the symplectic Floer homology of genus-2 surface diffeomorphisms, due to Ivan Smith. We use similar arguments at the end to extend our main result to infinitely many surgery slopes in the interval [3,5).

arXiv comment28 pages

References in corpus (21)

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• Khovanov homology detects the trefoils
• Instantons and odd Khovanov homology
• Instanton Floer homology and the Alexander polynomial
• Symplectic Floer homology of area-preserving surface diffeomorphisms
• Framed instanton homology and concordance
• Instantons and L-space surgeries
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• (1,1) L-space knots
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• Instanton Floer homology, sutures, and Euler characteristics
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Cited by in corpus (5)

• Fixed-point-free pseudo-Anosov homeomorphisms, knot Floer homology and the cinquefoil
• Knot surgery formulae for instanton Floer homology II: applications
• Torus knots, the A-polynomial, and SL(2,C)
• Instantons and Khovanov homology in ${\mathbb{RP}}^3$
• Small Heegaard genus and SU(2)