Due in class April 18.

p 155, section 2.2: 21, 22, 28a, 29

A. Suppose we have a smooth embedding \(f: S^1\times D^2 \hookrightarrow S^3\). Let \(K = f(S^1\times 0)\) and \(N(K) = f(S^1\times D^2)\). We call \(K\) a knot and \(N(K)\) its tubular neighborhood. We call \(S^3 – K\) the knot complement. Compute \(H_n(S^3 – K)\) for all \(n\) using Mayer-Vietoris. Carefully specify generators of all spaces involved.

B. Consider \(\mathbb{R}^3 – K \subset S^3 – K \). Using a Mayer-Vietoris sequence, find a generator for \(H_2(\mathbb{R}^3-K)\).

C. Write a short essay outlining a proof that for a homology theory \(h\) on the category of finite CW pairs, the group \(h_0(\ast)\) determines all homology groups \(h_n(X,A)\) for all CW pairs \((X,A)\). A flow chart may be useful.