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The Chain Complex of the Universal Cover of a Knot Complement

Every 3-manifold has a handle decomposition. Given a knot K : S →֒ S and a diagram for K we construct a handle decomposition for the complement X of K, a 3-dimensional manifold with boundary. We use this decomposition to extract the fundamental group of X and to write down the chain complex C∗(X̃) of the universal cover X̃. We then show how to obtain an algebraic formula approximating, on the chain level, the diagonal map ∆: X → X̃×π1(X)X̃. Taking account of the boundary ∂X, we can obtain the duality chain equivalence between the relative chain complex C∗(X̃, ∂̃X) and the dual complex C(X̃) . This encodes the geometric intersection and linking information of the manifold.