We discuss the linking structure of the attractor-repeller pairs in simple Smale flows on the 3-sphere in which the chaotic saddle set is modeled by four- band templates with twisted bands. We obtain new theorems which illustrate that the dynamics of simple Smale flows are sensitive to half-twists in the bands of the embedded template. Haynes and Sullivan showed that the attractor- repeller pair a∪r in a simple Smale flow with chaotic saddle set modeled by embedded template U^+ is either a Hopf link or a trefoil and meridian. By placing a single half-twist in a selected band of U^+, we obtain new templates that model chaotic saddle sets of Smale flows. For simple Smale flows on S^3 with chaotic saddle sets modeled by those templates, we find that such simple Smale flows are realizable and that a∪r must be a Hopf link, a figure-8 knot and meridian, or a trefoil and meridian. This is joint work with Michael Sullivan.