Deformations of dihedral representations

Heusener, Michael; Klassen, Eric

doi:10.1090/S0002-9939-97-04195-6

Deformations of dihedral representations
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by Michael Heusener and Eric Klassen PDF

Proc. Amer. Math. Soc. 125 (1997), 3039-3047 Request permission

Abstract:

G. Burde proved (1990) that the $\mathrm {SU}$ representation space of two-bridge knot groups is one-dimensional. The same holds for all torus knot groups. The aim of this note is to prove the following: Given a knot $k \subset S^3$ we denote by $\hat {C}_2$ its twofold branched covering space. Assume that there is a prime number $p$ such that $H_1(\hat {C}_2,\mathbb {Z}_p)\cong \mathbb {Z}_p$. Then there exist representations of the knot group onto the binary dihedral group $D_p \subset \mathrm {SU}$ and these representations are smooth points on a one-dimensional curve of representations into $\mathrm {SU}$.

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