This picture shows a three-branch spiral template. Four periodic orbits are drawn on its surface: a period-1 orbit (yellow) and three period-2 orbits (blue, green, and red).
If branches are labeled with distinct symbols, each orbit is uniquely identified by a symbolic itinerary listing the branches it successively visits. Here the yellow orbit is coded as "1" while the symbolic names of the blue, green and red orbits are "01", "02", and "12", respectively.
A template describes concisely the topological organization of periodic orbits. Any set of orbits projected onto the template has exactly the same knot invariants as in the original phase space. Computing these invariants is easy for template orbits. Look at the yellow and red orbits: you can see that they cross exactly twice. This means that each orbit winds once around the other, i.e., their linking number is 1. One of the key idea of topological analysis is that this knot invariant is completely determined by the stretching and squeezing mechanisms that organize the chaotic attractor, and thus that it provides a signature of these mechanisms.