Representations of knot groups in SU (2)

This thesis is a study of the structure of the space R(G) = Hom(G,SU(2)) of representations of integer knot and integer link groups into SU(2), where G is integer knot or integer link group. A representation is said to be a reducible representation if it is conjugate to a diagonal representation. The diagonal representations can be taken as the abelian representations and vice versa. Purpose of this work is to classify SU(2) representations of integer knot and link groups up to SO(3) equivalence. It is known that SO(3) = S^3/ {±I } acts on S^3 by conjugation. This action naturally extends on R(G) and since it is a free action when restricted to irreducible representations R^∗(G) one can define [R^∗(G)] = R^∗(G)/ action by SO(3). Throughout the thesis SU(2) and S^3 are regarded as isomorphic Lie groups and S^3 is a metric space with the metric d(x, y) = cos^−1 < x, y >∈ [0,π]. As in the complex numbers a unit quaternion Q is used in polar form Q = cosα+sinαq = e^{αq}. A SU(2) representation of the group can be thought of as a set of quaternions which satisfy corresponding relations in the group since generators of a knot group with Wirtinger presentation are the homotopy classes of meridians. Hence a representation can be regarded as a configuration of n points, number of generators of the knot group, in SU(2). So, being a subset of (SU(2))^n, the representation space inherits a (subspace) topology from (SU(2))^n. The compact-open topology can be assigned to this subset but subspace topology somehow more canonical although they are equivalent. For clarity it was discussed the knot case first and link is latter. The circle representations of integer knot and link groups have been classified before and taking this classification as a fundamental idea any SU(2) representation has been identified with a circle representation. Any SU(2) representation of integer knot or link group G induces a circle representation of G and conversely a circle representation can be lifted to an SU(2) representation. As a result the topology of representation space of G modulo SO(3) have been characterized. Keywords: Knot group, Link, Conjugate, Alexander polynomial, Representation