How to use the Reidemeister torsion

Firstly, we give the definition of the Reidemeister torsion, and explain basic properties, following V. G. Turaev [22], [23], [24]. Secondly, we consider the Reidemeister torsion of a homology lens space, which is the result of p/q-surgery along a knot K in a homology 3-sphere Σ. We denote the homology lens space by Σ(K; p/q). Main Theorem 1 is the case that K is a torus knot in S3. Main Theorem 2 is the case that the Alexander polynomial of K, ∆K(t), is degree 2. We judge when the homology lens spaces are homeomorphic to lens spaces by using the Reidemeister torsion. Statement of Main Theorems Some terms will be explained later. We state our two main theorems. Theorem (Moser [14]; Gordon [8]; Shimozawa [21]) Let Kr,s be the (r, s)-torus knot in S3, and M = S3(Kr,s; p/q) the result of p/q-surgery along Kr,s where |p|, |r|, |s | ≥ 2 and q = 0. Then there are three cases: (1) If |p−qrs | = 0, then M is a Seifert fibered space with three singular fibers of multiplicities |r|, |s | and |p − qrs|. In particular, (2) if |p − qrs | = 1, then M is the lens space L(p, qr2) (Figure 1). (3) If |p − qrs | = 0 (p/q = rs), then M is the connected sum of two lens spaces, L(r, s)L(s, r). We denote tha