Divisibility of twisted Alexander polynomials and fibered knots
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 1, p. 179-186
We prove that Wada’s twisted Alexander polynomial of a knot group associated to any nonabelian SL(2,𝔽)-representation is a polynomial. As a corollary, we show that it is always a monic polynomial of degree 4g-2 for a fibered knot of genus g.
Classification:  57M25,  57M05
@article{ASNSP_2005_5_4_1_179_0,
     author = {Kitano, Teruaki and Morifuji, Takayuki},
     title = {Divisibility of twisted Alexander polynomials and fibered knots},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 4},
     number = {1},
     year = {2005},
     pages = {179-186},
     zbl = {1117.57004},
     mrnumber = {2165406},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2005_5_4_1_179_0}
}
Kitano, Teruaki; Morifuji, Takayuki. Divisibility of twisted Alexander polynomials and fibered knots. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 1, pp. 179-186. http://www.numdam.org/item/ASNSP_2005_5_4_1_179_0/

[1] J. C. Cha, Fibred knots and twisted Alexander invariants, Trans. Amer. Math. Soc. 355 (2003), 4187-4200. | MR 1990582 | Zbl 1028.57004

[2] G. De Rham, Introduction aux polynomes d'un nœud, Enseign. Math. 13 (1968), 187-194. | Zbl 0157.54803

[3] H. Goda, T. Kitano and T. Morifuji, Reidemeister torsion, twisted Alexander polynomial and fibered knots, Comment. Math. Helv. 80 (2005), 51-61. | MR 2130565 | Zbl 1066.57008

[4] H. Goda and T. Morifuji, Twisted Alexander polynomial for SL(2,)-representations and fibered knots, C. R. Math. Acad. Sci. Soc. R. Can. 25 (2003), 97-101. | MR 2013157 | Zbl 1061.57012

[5] B. Jiang and S. Wang, Twisted topological invariants associated with representations, In: “Topics in Knot Theory”, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 399, Kluwer Academic Publishers, Dordrecht, 1993, 211-227. | MR 1257911 | Zbl 0815.55001

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

[7] P. Kirk and C. Livingston, Twisted knot polynomials: inversion, mutation and concordance, Topology 38 (1999), 663-671. | MR 1670424 | Zbl 0928.57006

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

[9] T. Kitano, M. Suzuki and M. Wada, Twisted Alexander polynomial and surjectivity of a group homomorphism, preprint. | MR 2171811

[10] K. Kodama, http://www.math.kobe-u.ac.jp/HOME/kodama/knot.html

[11] 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

[12] T. Morifuji, A twisted invariant for finitely presentable groups, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), 143-145. | MR 1801675 | Zbl 0988.57008

[13] T. Morifuji, Twisted Alexander polynomial for the braid group, Bull. Austral. Math. Soc. 64 (2001), 1-13. | MR 1848073 | Zbl 1008.20028

[14] L. Neuwirth, “Knot Groups”, Annals of Mathematics Studies, No. 56, Princeton University Press, Princeton, N.J., 1965. | MR 176462 | Zbl 0184.48903

[15] R. Riley, Nonabelian representations of 2-bridge knot groups, Quarterly J. Math. Oxford (2) 35 (1984), 191-208. | MR 745421 | Zbl 0549.57005

[16] M. Suzuki, Twisted Alexander polynomial for the Lawrence-Krammer representation, Bull. Austral. Math. Soc. 70 (2004), 67-71. | MR 2079361 | Zbl 1065.57013

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