At the end of Chapter 4, we saw how links could be decomposed into fragments called tangles. In particular, rational links have a decomposition into two trivial 2-tangles. In the last chapter we introduced a more rigid framework for studying 2-tangles using marked tangles. We now investigate the marked tangles that can be used to build rational links. This leads to a particularly nice classification scheme and a close relationship with the rational numbers, whence the name.

Generating rational tangles

A rational tangle is homeomorphic to the trivial 2-tangle. This means that it can be ‘unwound’ by sliding the endpoints around on the boundary sphere. We want to use this idea in reverse, starting with a basic object and applying a sequence of operations to it to build up complex-looking tangles.

Let us first define four operations that can be performed on a marked tangle (see Figure 8.1):

H Take the two right-hand ends (NE and SE) and twist them so that the over-crossing strand created has positive gradient.

V Take the two lower ends (SW and SE) and twist them so that the over-crossing strand created has positive gradient.

R The points NW and SE determine a line through the tangle. Using this line as an axis, rotate the tangle 180°.

F The points NW and SE determine a plane through the tangle that is orthogonal to the equatorial plane. Reflect the tangle in this plane. (Recall that the equatorial plane is the one containing the four end-points.)