The excellent hyperbolic structure that W. Thurston has now shown to exist on all suitable knot complements is still rather difficult to describe or write down explicitly in specific cases. There are published examples of this structure (exactly one knot and a large but restricted collection of links), but these all relied on the most exceptional feature that the rings R(θ) of the excellent p-reps concerned were always discrete rings of algebraic integers. In a companion paper we shall discuss methods of proving that an explicitly given discrete subgroup G of PSL(2, C) really is discrete when the definition of G did not make its discreteness clear. This situation often arises when the group is defined algebraically in relation to the solution of certain polynomial equations, and the matrix entries of G generate an algebraic number field of high degree. The methods of were developed specifically for the application to groups G of the type considered in this paper, and the present paper began life as a segment of, which thereby grew so large that it had to fission. We shall exhibit the excellent hyperbolic structure most explicitly for seven typical knots which exhibit the greatest range of behaviour that our small supply of worked examples permits.