We prove that the twisted Alexander polynomial of a torus knot with an irreducible -representation is locally constant. In the case of a 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.
@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}, pages = {395--406}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 11}, number = {2}, year = {2012}, zbl = {1255.57014}, mrnumber = {3011996}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2012_5_11_2_395_0/} }
TY - JOUR AU - Kitano, Teruaki AU - Morifuji, Takayuki TI - Twisted Alexander polynomials for irreducible $SL(2,\protect \mathbb{C})$-representations of torus knots JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2012 SP - 395 EP - 406 VL - 11 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2012_5_11_2_395_0/ LA - en ID - ASNSP_2012_5_11_2_395_0 ER -
%0 Journal Article %A Kitano, Teruaki %A Morifuji, Takayuki %T Twisted Alexander polynomials for irreducible $SL(2,\protect \mathbb{C})$-representations of torus knots %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2012 %P 395-406 %V 11 %N 2 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2012_5_11_2_395_0/ %G en %F 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, Série 5, Tome 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 | Zbl
[2] M. Hirasawa and K. Murasugi, Evaluations for the twisted Alexander polynomials of -bridge knots at , J. Knot Theory Ramifications 19 (2010), 1355–1400. | MR | Zbl
[3] D. Johnson, “A Geometric Form of Casson’s Invariant and its Connection to Reidemeister Torsion”, unpublished Lecture Notes.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 | Zbl
[5] T. Kitano, Reidemeister torsion of Seifert fibered spaces for -representations, Tokyo J. Math. 17 (1994), 59–75. | MR | Zbl
[6] T. Kitano, Twisted Alexander polynomial and Reidemeister torsion, Pacific J. Math. 174 (1996), 431–442. | MR | Zbl
[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. | EuDML | Numdam | MR | Zbl
[8] T. Kitayama, Normalization of twisted Alexander invariants, arXiv:0705.2371. | MR
[9] X. S. Lin, Representations of knot groups and twisted Alexander polynomials, Acta Math. Sin. (Engl. Ser.) 17 (2001), 361–380. | MR | Zbl
[10] T. Morifuji, Twisted Alexander polynomials of twist knots for nonabelian representations, Bull. Sci. Math. 132 (2008), 439–453. | MR | Zbl
[11] D. Silver and S. Williams, Dynamics of twisted Alexander invariants, Topology Appl. 156 (2009), 2795–2811. | MR | Zbl
[12] M. Wada, Twisted Alexander polynomial for finitely presentable groups, Topology 33 (1994), 241–256. | MR | Zbl