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Metabelian SL(n, $\mathbb C$ ) representations of knot groups IV: twisted Alexander polynomials

Published online by Cambridge University Press:  20 September 2013

HANS U. BODEN
Affiliation:
Mathematics & Statistics, McMaster University, Hamilton, Ontario, L8S-4K1, Canada. e-mail: boden@mcmaster.ca
STEFAN FRIEDL
Affiliation:
Mathematisches Institut, Universität zu Köln, 50931 Köln, Germany e-mail: sfriedl@gmail.com
Corresponding

Abstract

In this paper we will study properties of twisted Alexander polynomials of knots corresponding to metabelian representations. In particular we answer a question of Wada about the twisted Alexander polynomial associated to the tensor product of two representations, and we settle several conjectures of Hirasawa and Murasugi.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2013 

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References

[1]Boden, H. U. and Friedl, S.Metabelian SL(n, ${\mathbb C}$) representations of knot groups. Pacific J. Math. 238 (2008), 725.CrossRefGoogle Scholar
[2]Boden, H. U. and Friedl, S.Metabelian SL(n, ${\mathbb C}$) representations of knot groups II: fixed points. Pacific J. Math. 249 (2011), 110.CrossRefGoogle Scholar
[3]Boden, H. U. and Friedl, S. Metabelian SL(n, $\mathbb C$) representations of knot groups III: deformations. Preprint (2012), math.GT 1208.1708. To appear in Q. J. Math.Google Scholar
[4]Cha, J. C. and Friedl, S.Twisted torsion invariants and link concordance. Forum Math. 25 (2013), 471504.Google Scholar
[5]Cha, J. C. and Livingston, C. KnotInfo: table of knot invariants. Online at http://www.indiana.edu/~knotinfo (September 1, 2012).Google Scholar
[6]Friedl, S.Eta invariants as sliceness obstructions and their relation to Casson–Gordon invariants. Alg. Geom. Topol. 4 (2004), 893934.CrossRefGoogle Scholar
[7]Friedl, S. KnotTwister (2012), http://www.mi.uni-koeln.de/~stfriedl.Google Scholar
[8]Friedl, S. and Vidussi, S.A survey of twisted Alexander polynomials. The Mathematics of Knots. 45–94, Contrib. Math. Comput. Sci., 1 (Springer, Heidelberg, 2011).Google Scholar
[9]Herald, C., Kirk, P. and Livingston, C.Metabelian representations, twisted Alexander polynomials, knot slicing and mutation. Math. Zeit. 265 (2010), 925949.CrossRefGoogle Scholar
[10]Hirasawa, M. and Murasugi, K. Twisted Alexander polynomials of 2-bridge knots associated to metabelian representations, 2009 preprint math.GT 0903.1689.Google Scholar
[11]Hirasawa, M. and Murasugi, K. Twisted Alexander polynomials of 2-bridge knots associated to metacyclic representations. Preprint (2009) math.GT 0903.0147.Google Scholar
[12]Hoste, J. and Shanahan, P.Twisted Alexander polynomials of 2-bridge knots. J. Knot Theory Ramifications 22, no. 1 (2013), 129.CrossRefGoogle Scholar
[13]Kirk, P. and Livingston, C.Twisted Alexander invariants, Reidemeister torsion and Casson–Gordon invariants. Topology 38, no. 3 (1999), 635661.CrossRefGoogle Scholar
[14]Kitano, T.Twisted Alexander polynomials and Reidemeister torsion. Pacific J. Math. 174, no. 2 (1996), 431442.CrossRefGoogle Scholar
[15]Lang, S.Algebra. (3rd revised ed.). Graduate Texts in Mathematics, 211 (Springer-Verlag, New York, 2002).CrossRefGoogle Scholar
[16]Lin, X. S.Representations of knot groups and twisted Alexander polynomials. Acta Math. Sin. (Engl. Ser.) 17, no. 3 (2001), 361380.CrossRefGoogle Scholar
[17]Stein, W. A.et al. Sage Mathematics Software (Version 4ċ8), (The Sage Development Team, 2012), http://www.sagemath.orgGoogle Scholar
[18]Stoimenov, A. A table of braid descriptions of knots through 12 crossings (2012), http://stoimenov.net/stoimeno/homepage/ptab/braid.outGoogle Scholar
[19]Wada, M.Twisted Alexander polynomial for finitely presentable groups. Topology 33, no. 2 (1994), 241256.CrossRefGoogle Scholar
[20]Wada, M.Twisted Alexander polynomial revisited. RIMS Kôkyûroku 1747 (2011), 140144.Google Scholar

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