Twisted Alexander polynomials for irreducible SL(2,)-representations of torus knots
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 2, p. 395-406

We prove that the twisted Alexander polynomial of a torus knot with an irreducible SL(2,)-representation is locally constant. In the case of a (2,q) torus knot, we can give an explicit formula for the twisted Alexander polynomial and deduce Hirasawa-Murasugi’s formula for the total twisted Alexander polynomial. We also give examples which address a mis-statement in a paper of Silver and Williams.

Published online : 2018-06-21
Classification:  57M27
@article{ASNSP_2012_5_11_2_395_0,
     author = {Kitano, Teruaki and Morifuji, Takayuki},
     title = {Twisted Alexander polynomials for irreducible $SL(2,\protect \mathbb{C})$-representations of torus knots},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 11},
     number = {2},
     year = {2012},
     pages = {395-406},
     mrnumber = {3011996},
     zbl = {1255.57014},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2012_5_11_2_395_0}
}
Kitano, Teruaki; Morifuji, Takayuki. Twisted Alexander polynomials for irreducible $SL(2,\protect \mathbb{C})$-representations of torus knots. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 2, pp. 395-406. http://www.numdam.org/item/ASNSP_2012_5_11_2_395_0/

[1] R. H. Crowell and R. H. Fox, “Introduction to Knot Theory”, Grad. Texts Math., Vol. 57, Springer-Verlag, 1977. | MR 445489 | Zbl 0362.55001

[2] M. Hirasawa and K. Murasugi, Evaluations for the twisted Alexander polynomials of 2-bridge knots at ±1, J. Knot Theory Ramifications 19 (2010), 1355–1400. | MR 2735813 | Zbl 1207.57019

[3] D. Johnson, “A Geometric Form of Casson’s Invariant and its Connection to Reidemeister Torsion”, unpublished Lecture Notes.

[4] P. Kirk and C. Livingston, Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants, Topology 38 (1999), 635–661. | MR 1670420 | Zbl 0928.57005

[5] T. Kitano, Reidemeister torsion of Seifert fibered spaces for SL(2;)-representations, Tokyo J. Math. 17 (1994), 59–75. | MR 1279569 | Zbl 0846.55010

[6] T. Kitano, Twisted Alexander polynomial and Reidemeister torsion, Pacific J. Math. 174 (1996), 431–442. | MR 1405595 | Zbl 0863.57001

[7] T. Kitano and T. Morifuji, Divisibility of twisted Alexander polynomials and fibered knots, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), 179–186. | Numdam | MR 2165406 | Zbl 1117.57004

[8] T. Kitayama, Normalization of twisted Alexander invariants, arXiv:0705.2371. | MR 3401962

[9] X. S. Lin, Representations of knot groups and twisted Alexander polynomials, Acta Math. Sin. (Engl. Ser.) 17 (2001), 361–380. | MR 1852950 | Zbl 0986.57003

[10] T. Morifuji, Twisted Alexander polynomials of twist knots for nonabelian representations, Bull. Sci. Math. 132 (2008), 439–453. | MR 2426646 | Zbl 1147.57018

[11] D. Silver and S. Williams, Dynamics of twisted Alexander invariants, Topology Appl. 156 (2009), 2795–2811. | MR 2556037 | Zbl 1200.57006

[12] M. Wada, Twisted Alexander polynomial for finitely presentable groups, Topology 33 (1994), 241–256. | MR 1273784 | Zbl 0822.57006