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Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants. (English) Zbl 0928.57005
In this and a subsequent paper [ibid., 663-671 (1999; Zbl 0928.57006), see the review below] the authors provide a substantial account of the construction and properties of twisted Alexander invariants and give some interesting applications. The main definition given here applies to a finite complex \(X\) with fundamental group \(\pi\), together with a homomorphism \(\varepsilon: \pi\to \mathbb{Z}\) and a representation \(\rho:\pi \to\text{GL}(V)\) on some finite dimensional vector space \(V\) over \(F\). The homology of the infinite cyclic cover of \(X\) twisted by \(\rho\) can then be regarded as a module over \(F[\mathbb{Z}]\), leading to the definition of the twisted Alexander polynomials as elements of \(F[\mathbb{Z}]\) up to a unit. The classical Alexander polynomial of a knot with group \(\pi\) arises in this context when \(\rho\) is the trivial representation. The construction is interpreted in the general framework of Reidemeister torsion. The authors show how the classical Fox calculus can be used in the calculations, and how the earlier work of X.-S. Lin [Representations of knot groups and twisted Alexander polynomials, Preprint, 1990] and M. Wada [Topology 33, No. 2, 241-256 (1994; Zbl 0822.57006)] on twisted Alexander invariants for knots can be seen in the current framework. They describe the general case of knots in \(S^3\) and develop ‘Mayer-Vietoris’ features of the invariants which are used to give a nice account of the behaviour for satellite knots. They then go on to discuss symmetries of the invariants, extending Milnor’s work on Reidemeister torsion and duality to the twisted case. The final topic treated is that of obstruction to slicing, where the use of the twisted Alexander invariant on prime-power covers related to a knot gives a connection with its Casson-Gordon invariants.

57M25 Knots and links in the \(3\)-sphere (MSC2010)
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