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Representations of knot groups and twisted Alexander polynomials. (English) Zbl 0986.57003

The author considers knots in the 3-sphere and presents a twisted version of the Alexander polynomial associated with a matrix representation of the knot group. The classical Alexander polynomial is the one associated with 1-dimensional representations. Examples of knots with the same classical Alexander polynomial but different twisted Alexander polynomials are given.
This article appeared first in 1990 as a Columbia University preprint. Since then further works on the same topic have been published by several authors. See for example [P. Kirk and C. Livingston, Topology 38, No. 3, 635-661 (1999; Zbl 0928.57005)].

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)

Citations:

Zbl 0928.57005
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References:

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[2] S. Cappell, R. Lee, E. Miller, A symplectic geometry approach to generalized Casson’s invariants of 3- manifold, Bull. A. M. S., 1990, 22:269–275 · Zbl 0699.57009 · doi:10.1090/S0273-0979-1990-15885-9
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[5] T. Kitano, Twisted Alexander polynomial and Reidemeister torsion, Pacific J. Math., 1996, 174:431–442 · Zbl 0863.57001
[6] P. Kirk, C. Livingston, Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants, Topology, 1999, 38:635–661 · Zbl 0928.57005 · doi:10.1016/S0040-9383(98)00039-1
[7] P. Kirk, C. Livingston, Twisted knot polynomials:inversion, mutation and concordance, Topology, 1999, 38:663–671 · Zbl 0928.57006 · doi:10.1016/S0040-9383(98)00040-8
[8] C. McA. Gordon, Some aspects of classical knot theory, Lecture Notes in Math., 685:1–60 · Zbl 0386.57002
[9] K. Reidemeister, Knotentheorie, New York:Chelsea Publ. Co., 1948
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[11] R. H. Fox, A quick trip through knot theory, Topology of manifolds, Prentice Hall, 1960, 120–167 · Zbl 1246.57002
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