In this chapter we give space a book-like infrastructure of pages attached to a binding. This enables us to study links in two complementary ways: aligned within the pages or running transversely to them. This provides two new geometric link invariants, lower bounds for which can be obtained from the 2-variable link polynomials F(a, x) and P(v, z).

First, then, let us define the infrastructure. Think of ℝ3 as ℂ × ℝ with coordinate system (r, θ, z). The z-axis will be the binding for the book. Using polar coordinates in the complex plane rather than the standard cartesian ones makes it simpler to describe the pages. For a fixed value of θ the set Hθ = {(r, θ, z) ∈ ℝ3} is the half-plane at angle θ. As θ ranges from 0 to 2π these half-planes fill space – they are the pages of an open-book decomposition of ℝ3.

If we add the point at infinity to form S3 = ℝ3 ∪ {∞} then the binding becomes the circle z-axis ∪ {∞} and the pages become discs. In both ℝ3 and S3, if we remove the binding, we are left with a space homeomorphic to an open solid torus, and the pages are just the meridional discs.

Braid presentations

In §7.10 we met a special form of n-tangle, called a braid, in which all the strings descend monotonically.