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TWISTED ALEXANDER POLYNOMIALS AND CHARACTER VARIETIES OF 2-BRIDGE KNOT GROUPS

TAEHEE KIM

Department of Mathematics, Konkuk University, Seoul 143-701, Republic of Korea

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and

TAKAYUKI MORIFUJI

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

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https://doi.org/10.1142/S0129167X11007653Cited by:7 (Source: Crossref)

Abstract

We study the twisted Alexander polynomial from the viewpoint of the SL(2, ℂ)-character variety of nonabelian representations of a knot group. It is known that if a knot is fibered, then the twisted Alexander polynomials associated with nonabelian SL(2, ℂ)-representations are all monic. In this paper, we show that for a 2-bridge knot there exists a curve component in the SL(2, ℂ)-character variety such that if the knot is not fibered then there are only finitely many characters in the component for which the associated twisted Alexander polynomials are monic. We also show that for a 2-bridge knot of genus g, in the above curve component for all but finitely many characters the associated twisted Alexander polynomials have degree 4g - 2.

Keywords:

AMSC: 57M27, 57M05, 57M25

References

M. Boileauet al., Comm. Anal. Geom. 18, 121 (2010). Crossref, Web of Science, Google Scholar
G. Burde, Math. Ann. 173, 24 (1967). Crossref, Web of Science, Google Scholar
J. C. Cha, Trans. Amer. Math. Soc. 355, 4187 (2003). Crossref, Web of Science, Google Scholar
R. Crowell, Ann. of Math. (2) 69, 258 (1959). Crossref, Web of Science, Google Scholar
M. Culler and P. B. Shalen, Ann. of Math. 117, 109 (1983). Crossref, Web of Science, Google Scholar
M. Culler and P. B. Shalen, Invent. Math. 75, 537 (1984). Crossref, Web of Science, Google Scholar
G. de Rham, Enseign. Math. (2) 13, 187 (1967). Web of Science, Google Scholar
N. Dunfield, S. Friedl and N. Jackson, Twisted Alexander polynomials of hyperbolic knots, to appear in Experiment. Math . Google Scholar
R. H. Fox, Ann. of Math. 57, 547 (1953). Crossref, Web of Science, Google Scholar
S. Friedl and T. Kim, Topology 45, 929 (2006). Crossref, Web of Science, Google Scholar
S. Friedl, T. Kim and T. Kitayama, Poincaré duality and degrees of twisted Alexander polynomials, to appear in Indiana Univ. Math. J . Google Scholar
S. Friedl and S. Vidussi, Ann. of Math. 173, 1587 (2011). Crossref, Web of Science, Google Scholar
S. Friedl and S. Vidussi, The Mathematics of Knots: Theory and Application, Contributions in Mathematical and Computational Sciences, eds. M. Banagl and D. Vogel (Springer, 2010) pp. 45–94. Google Scholar
H. Goda, T. Kitano and T. Morifuji, Comment. Math. Helv. 80, 51 (2005). Web of Science, Google Scholar
H. Goda and T. Morifuji, C. R. Math. Acad. Sci. Soc. R. Can. 25, 97 (2003). Google Scholar
J. Hillman, D. Silver and S. Williams, Algebr. Geom. Topol. 10, 1017 (2010). Crossref, Web of Science, Google Scholar
M. Hirasawa and K. Murasugi, Twisted Alexander polynomials of 2-bridge knots associated to metacyclic representations, preprint (2009) , arXiv:0903.0147 . Google Scholar
J. Hoste and P. D. Shanahan, J. Knot Theory Ramifications 13, 193 (2004). Link, Web of Science, Google Scholar
P. Kirk and C. Livingston, Topology 38, 635 (1999). Crossref, Web of Science, Google Scholar
T. Kitano, Pacific J. Math. 174, 431 (1996). Crossref, Web of Science, Google Scholar
T. Kitano and T. Morifuji, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 4, 179 (2005). Google Scholar
P. B. Kronheimer and T. S. Mrowka, Math. Res. Lett. 11, 741 (2004). Crossref, Web of Science, Google Scholar
T. T. Q. Le, Mat. Sb. 184, 57 (1993). Google Scholar
X. S. Lin, Acta Math. Sin. Engl. Ser. 17, 361 (2001). Crossref, Web of Science, Google Scholar
M. L. Macasieb, K. L. Petersen and R. M. Van Luijk, Proc. London Math. Soc. (3) 103, 473 (2011). Crossref, Web of Science, Google Scholar
T. Morifuji, Bull. Sci. Math. 132, 439 (2008). Crossref, Web of Science, Google Scholar
K. Murasugi, J. Math. Soc. Japan 10, 94 (1958). Crossref, Web of Science, Google Scholar
K. Murasugi, J. Math. Soc. Japan 10, 235 (1958). Crossref, Web of Science, Google Scholar
K. Murasugi, Amer. J. Math. 85, 544 (1963). Crossref, Web of Science, Google Scholar
K. Murasugi, Contemp. Math. 416, 167 (2006). Crossref, Google Scholar
F. Nagasato and Y. Yamaguchi, On the geometry of the slice of trace-free SL(2, ℂ)-characters of a knot group, to appear in Math. Ann . Google Scholar
T. Ohtsuki, J. Math. Soc. Japan 46, 51 (1994). Crossref, Web of Science, Google Scholar
T. Ohtsuki, R. Riley and M. Sakuma, The Zieschang Gedenkschrift, Geometry and Topology Monographs 14 (Mathematical Sciences Publishers, 2008) pp. 417–450. Crossref, Google Scholar
R. Riley, Q. J. Math. Ser. (2) 35, 191 (1984). Crossref, Web of Science, Google Scholar
M. Wada, Topology 33, 241 (1994). Crossref, Web of Science, Google Scholar