On reciprocality of twisted Alexander invariants. (English) Zbl 1200.57005

By a classical result, the Alexander polynomial of a knot is reciprocal, due to duality of presentations or spaces, depending of one’s point of view. In the presence of an \(n\)-dimensional linear representation \(\gamma : \pi \rightarrow GL_nR\) of the knot group \(\pi\) we have twisted polynomial invariants, and the paper goes into the question of reciprocality of those. By Theorem 3.2, the twisted Reidemeister torsion \(\tau_\gamma(t)\) and the twisted Alexander polynomial \(\Delta_\gamma(t)\) are reciprocal, provided \(R={\mathbb C}\) and \(\gamma\) is conjugate to its dual (inverse-transpose). For \(R={\mathbb Z}\), a representation into \(SL_3{\mathbb Z}\) is presented such that \(\tau_\gamma(t)\) is non-reciprocal.


57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
Full Text: DOI arXiv


[1] J C Cha, Fibred knots and twisted Alexander invariants, Trans. Amer. Math. Soc. 355 (2003) 4187 · Zbl 1028.57004
[2] S Friedl, S Vidussi, Twisted Alexander polynomials and symplectic structures, Amer. J. Math. 130 (2008) 455 · Zbl 1154.57021
[3] S Friedl, S Vidussi, Twisted Alexander polynomials, symplectic 4-manifolds and surfaces of minimal complexity, Banach Center Publ. 85, Polish Acad. Sci. Inst. Math., Warsaw (2009) 43 · Zbl 1170.57019
[4] J A Hillman, C Livingston, S Naik, Twisted Alexander polynomials of periodic knots, Algebr. Geom. Topol. 6 (2006) 145 · Zbl 1097.57010
[5] A Kawauchi, A survey of knot theory, Birkhäuser Verlag (1996) · Zbl 0861.57001
[6] P Kirk, C Livingston, Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants, Topology 38 (1999) 635 · Zbl 0928.57005
[7] T Kitano, Twisted Alexander polynomial and Reidemeister torsion, Pacific J. Math. 174 (1996) 431 · Zbl 0863.57001
[8] T Kitayama, Refinement of twisted Alexander invarariants and sign-determined Reidemeister torsions
[9] I Kovacs, D S Silver, S G Williams, Determinants of commuting-block matrices, Amer. Math. Monthly 106 (1999) 950 · Zbl 0981.15005
[10] X S Lin, Representations of knot groups and twisted Alexander polynomials, Acta Math. Sin. \((\)Engl. Ser.\()\) 17 (2001) 361 · Zbl 0986.57003
[11] J Milnor, Two complexes which are homeomorphic but combinatorially distinct, Ann. of Math. \((2)\) 74 (1961) 575 · Zbl 0102.38103
[12] J Milnor, A duality theorem for Reidemeister torsion, Ann. of Math. \((2)\) 76 (1962) 137 · Zbl 0108.36502
[13] D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish (1976) · Zbl 0339.55004
[14] D S Silver, S G Williams, Dynamics of twisted Alexander invariants, Topology Appl. 156 (2009) 2795 · Zbl 1200.57006
[15] G Torres, R H Fox, Dual presentations of the group of a knot, Ann. of Math. \((2)\) 59 (1954) 211 · Zbl 0055.16805
[16] M Wada, Twisted Alexander polynomial for finitely presentable groups, Topology 33 (1994) 241 · Zbl 0822.57006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.