SU(2)-cyclic surgeries and the pillowcase

Abstract

We study knots in $S^3$ with infinitely many $SU(2)$-cyclic surgeries, which are Dehn surgeries such that every representation of the resulting fundamental group into $SU(2)$ has cyclic image. We show that for every such nontrivial knot $K$, its set of $SU(2)$-cyclic slopes is bounded and has a unique limit point, which is both a rational number and a boundary slope for $K$. We also show that such knots are prime and have infinitely many instanton L-space surgeries. Our methods include the application of holonomy perturbation techniques to instanton knot homology, using a strengthening of recent work by the second author.

66 pages, 6 figures; v2: revised following referee reports

References in corpus (3)

• Survey article: Seventy years of Salem numbers
• Dehn surgery, the fundamental group and SU(2)
• Instantons and L-space surgeries

Cited by in corpus (5)

• Instantons and L-space surgeries
• Small Dehn surgery and SU(2)
• SU(2) representations and a large surgery formula
• Torus knots, the A-polynomial, and SL(2,C)
• Surgery obstructions and character varieties