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Twisted Alexander polynomials of torus links

Department of Information Systems Science, Faculty of Science and Engineering, Soka University, 1-236, Tangi-Cho, Hachioji, Tokyo 192-8577, Japan

E-mail Address: kitano@soka.ac.jp

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Takayuki Morifuji

Department of Mathematics, Keio University, Hiyoshi Campus, Yokohama 223-8521, Japan

E-mail Address: morifuji@z8.keio.jp

Corresponding author.

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, and

Anh T. Tran

Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX 75080, USA

E-mail Address: att140830@utdallas.edu

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https://doi.org/10.1142/S0218216520500169Cited by:0 (Source: Crossref)

Abstract

In this paper, we give an explicit formula for the twisted Alexander polynomial of any torus link and show that it is a locally constant function on the $SL (2, ℂ)$ -character variety. We also discuss similar things for the higher-dimensional twisted Alexander polynomial and the Reidemeister torsion.

Keywords:

AMSC: 57M27, 57M25

References

1. M. Culler and P. B. Shalen, Varieties of group representations and splittings of 3-manifolds, Ann. Math. 117 (1983) 109–146. Crossref, Web of Science, Google Scholar
2. S. Friedl and S. Vidussi, A survey of twisted Alexander polynomials, in The Mathematics of Knots: Theory and Application, eds. M. Banagl and D. Vogel (Contributions in Mathematical and Computational Sciences, Springer, Heidelberg, 2010), pp. 45–94. Google Scholar
3. H. Goda and T. Morifuji, Twisted Alexander polynomial for $SL (2, ℂ)$ -representations and fibered knots, C. R. Math. Acad. Sci. Soc. R. Can. 25 (2003) 97–101. Google Scholar
4. F. Gonzárez-Acuña and J. M. Montesinos-Amiliba, On the character variety of group representations in $SL (2, ℂ)$ and $PSL (2, ℂ)$ , Math. Z. 214 (1993) 627–652. Crossref, Web of Science, Google Scholar
5. D. Johnson, A geometric form of Casson’s invariant and its connection to Reidemeister torsion, unpublished lecture notes. Google Scholar
6. T. Kitano, Twisted Alexander polynomial and Reidemeister torsion, Pacific J. Math. 174 (1996) 431–442. Crossref, Web of Science, Google Scholar
7. T. Kitano, Introduction to twisted Alexander polynomials and related topics, Winter Braids Lect. Notes, Vol. 2 (2015), Winter Braids V (Mathdoc, Pau, 2015), Exp. No. 4, 35 pp. Google Scholar
8. 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. Google Scholar
9. T. Kitano and T. Morifuji, Twisted Alexander polynomials for irreducible $SL (2, ℂ)$ -representations of torus knots, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (2012) 395–406. Google Scholar
10. X.-S. Lin, Representations of knot groups and twisted Alexander polynomials, Acta Math. Sin. (Engl. Ser.) 17 (2001) 361–380. Crossref, Web of Science, Google Scholar
11. J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966) 348–426. Crossref, Web of Science, Google Scholar
12. T. Morifuji, Representations of knot groups into $SL (2, ℂ)$ and twisted Alexander polynomials, Handbook of Group Actions, Volume I, Advanced Lectures Mathematics (ALM), Vol. 31 (International Press, Somerville, MA, 2015), pp. 527–576. Google Scholar
13. V. Muñoz, The $SL (2, ℂ)$ -character varieties of torus knots, Rev. Mat. Complut. 22 (2009) 489–497. Crossref, Web of Science, Google Scholar
14. M. Rost and H. Zieschang, Meridional generators and plat presentations of torus links, J. London Math. Soc. (2) 35 (1987) 551–562. Crossref, Web of Science, Google Scholar
15. M. Wada, Twisted Alexander polynomial for finitely presentable groups, Topology 33 (1994) 241–256. Crossref, Web of Science, Google Scholar
16. Y. Yamaguchi, Higher even dimensional Reidemeister torsion for torus knot exteriors, Math. Proc. Cambridge Philos. Soc. 155 (2013) 297–305. Crossref, Web of Science, Google Scholar
17. Y. Yamaguchi, A surgery formula for the asymptotics of the higher-dimensional Reidemeister torsion and Seifert fibered spaces, Indiana Univ. Math. J. 66 (2017) 463–493. Crossref, Web of Science, Google Scholar