Middle East Technical University Fall 2002 Math 744 Fundamental Techniques in Differential Topology

PROBLEM LIST V

Alexander Polynomial and the Torsion

Problem 1. Consider a real vector space V as a Λ-module, Λ = ℝ[t, t^{−1}], where t act on V as an invertible linear operator T : V → V. Show that

(1) V is a torsion module (that is contains no free summands over Λ);

(2) the torsion |V |_{Λ} coincides with the characteristic polynomial, Δ_{L}, of L.

Problem 2.

(1) Find a Seifert matrix, S, and the Alexander polynomial Δ_{K} of a torus (2, 2n+1)- knot K.

(2) Compare your result with the characteristic polynomial of the monodromy matrix H = S^{T} S^{−1} for this knot.

(3) Represent H as a product of the Dehn twists around the generators of the Seifert surface.

Problem 3. Consider the mapping torus M_{f} of a homeomorphism f: T^{2} → T^{2}, where f = t_{a}t_{b} is a product of Dehn twists around the generators a, b of the torus T^{2}.

(1) Describe generators of C_{∗} = C_{∗}(M̃_{f}) over Λ = ℤ[t, t^{−1}] (specifying the corresponding cells in M̃_{f}).

(2) Describe the differentials in the complex C_{∗}.

(3) Describe the complex C_{∗}(M_{f}; ϕ), where ϕ: Λ → ℂ is a representation mapping t into z ∈ ℂ^{∗}.

(4) Determine for which values of z the homology H_{∗}(M_{f}, ϕ) vanish.

(5) Determine the Reidemeister torsion for those values z.

(6) Observe the relation between the Reidemeister torsion and the Alexander polynomial in this example.

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