Classifying the types of G-symmetry on knots.

If G is a finite group, we define a G-symmetric knot (K, alpha) to consist of a faithful action alpha of G on S^3 and a G-equivariant smooth embedding K of S^1 in S^3 (for some action of G on S^1). We define the type of a G-symmetric knot (K, alpha) to consist of whatever can be determined from the action of G on the oriented smooth manifolds S^3 and on S^1, ignoring the specifics of the knot K. Having made this definition, I will show that the analysis of types reduces largely to orthogonal representation theory of cyclic and dihedral groups. From here, I will explain which types arise from G-symmetric structures on prime knots.