By J. L. Dupont, I. H. Madsen
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Additional info for Algebraic Topology, Aarhus 1978
Tm 5. Finally given a link L we also drop the representation from the notation when the representation is the trivial representation to GL(1, Z). With all these conventions, given a knot K ⊂ S 3 , the polynomial Z[t ±1 ] is just the ordinary Alexander polynomial. 5 Computation of Twisted Alexander Polynomials Let N be a 3-manifold with empty or toroidal boundary, α : π1 (N ) → GL(k, R[F ]) a representation with R a Noetherian UFD and F a free abelian group. Given a finite presentation for π1 (N ) the polynomials αN,1 ∈ R[F ] and αN,0 ∈ R[F ] can be computed efficiently using Fox calculus (cf.
Theorem 7 Let i : X n → M n+2 , n ≡ 0(4), be a PL embedding of a compact oriented PL pseudomanifold X in a closed oriented PL manifold M which induces a stratification of the form X = Xn ⊃ X4 ⊃ X2 ⊃ X−1 = ∅, such that (i) for every connected component V of X4 − X2 , the closure V is a 4-manifold, (ii) the link pair of every such V is a (necessarily nontrivial but locally flat) spherical knot (S n−3 , S n−5 ) with definite real Blanchfield form of rank rV , (iii) X2 is a disjoint union of 2-spheres, and (iv) for every such S 2 and 4-dimensional V with S 2 ⊂ V , the latter embedding is locally flat with zero self-intersection number.
In Sect. 3 we discuss basic properties of twisted invariants, in particular we discuss the relationship between twisted Reidemeister torsion and twisted Alexander polynomials and we discuss the effect of Poincaré duality on twisted invariants. 4 contains applications to distinguishing knots and links using twisted invariants. In Sect. 5 we outline the results of Kirk and Livingston regarding the behavior of twisted invariants under knot concordance and we extend the results to the study of doubly slice knots and ribbon knots.
Algebraic Topology, Aarhus 1978 by J. L. Dupont, I. H. Madsen