Representations of knot groups in 𝑆𝑈(2)

Abstract

This paper is a study of the structure of the space R ( K ) R(K) of representations of classical knot groups into SU ( 2 ) {\text {SU}}(2) . Let R ^ ( K ) \hat R(K) equal the set of conjugacy classes of irreducible representations. In § I \S I , we interpret the relations in a presentation of the knot group in terms of the geometry of SU ( 2 ) {\text {SU}}(2) ; using this technique we calculate R ^ ( K ) \hat R(K) for K K equal to the torus knots, twist knots, and the Whitehead link. We also determine a formula for the number of binary dihedral representations of an arbitrary knot group. We prove, using techniques introduced by Culler and Shalen, that if the dimension of R ^ ( K ) \hat R(K) is greater than 1 1 , then the complement in S 3 {S^3} of a tubular neighborhood of K K contains closed, nonboundary parallel, incompressible surfaces. We also show how, for certain nonprime and doubled knots, R ^ ( K ) \hat R(K) has dimension greater than one. In § I I \S II , we calculate the Zariski tangent space, T ρ ( R ( K ) ) {T_\rho }(R(K)) , for an arbitrary knot K K , at a reducible representation ρ \rho , using a technique due to Weil. We prove that for all but a finite number of the reducible representations, dim ⁡ T ρ ( R ( K ) ) = 3 \dim {T_\rho }(R(K))= 3 . These nonexceptional representations possess neighborhoods in R ( K ) R(K) containing only reducible representations. At the exceptional representations, which correspond to real roots of the Alexander polynomial, dim ⁡ T ρ ( R ( K ) ) = 3 + 2 k \dim {T_\rho }(R(K)) = 3 + 2k for a positive integer k k . In those examples analyzed in this paper, these exceptional representations can be expressed as limits of arcs of irreducible representations. We also give an interpretation of these "extra" tangent vectors as representations in the group of Euclidean isometries of the plane.