No Access

ON THE TWISTED ALEXANDER POLYNOMIAL FOR REPRESENTATIONS INTO SL₂(ℂ)

ANH T. TRAN

Department of Mathematics, The Ohio State University Columbus, OH 43210, USA

https://doi.org/10.1142/S0218216513500594Cited by:1 (Source: Crossref)

Abstract

We study the twisted Alexander polynomial Δ_K,ρ of a knot K associated to a non-abelian representation ρ of the knot group into SL₂(ℂ). It is known for every knot K that if K is fibered, then for every non-abelian representation, Δ_K,ρ is monic and has degree 4g(K) – 2 where g(K) is the genus of K. Kim and Morifuji recently proved the converse for 2-bridge knots. In fact they proved a stronger result: if a 2-bridge knot K is non-fibered, then all but finitely many non-abelian representations on some component have Δ_K,ρ non-monic and degree 4g(K) – 2. In this paper, we consider two special families of non-fibered 2-bridge knots including twist knots. For these families, we calculate the number of non-abelian representations where Δ_K,ρ is monic and calculate the number of non-abelian representations where the degree of Δ_K,ρ is less than 4g(K) – 2.

Keywords:

AMSC: 57M27, 57M25

References

G. Burde and H. Zieschang , Knots , de Gruyter Studies in Mathematics 5 ( de Gruyter , Berlin , 2003 ) . Google Scholar
R. Crowell, Ann. Math. 69(2), 258 (1959), DOI: 10.2307/1970181. Web of Science, Google Scholar
N. Dunfield, S. Friedl and N. Jackson, Exp. Math. 21(4), 329 (2012), DOI: 10.1080/10586458.2012.669268. Web of Science, Google Scholar
S. Friedl and T. Kim, Topology 45, 929 (2006), DOI: 10.1016/j.top.2006.06.003. Web of Science, Google Scholar
S. Friedl and S. Vidussi, The Mathematics of Knots: Theory and Application, Contrib. Math. Comput. Sci 1, eds. M. Banagl and D. Vogel (2011) pp. 45–94. Google Scholar
S. Friedl and S. Vidussi, Ann. Math. 173, 1587 (2011), DOI: 10.4007/annals.2011.173.3.8. Web of Science, Google Scholar
S. Friedl and S. Vidussi, The Thurston norm and twisted Alexander polynomials, preprint (2012) , arXiv:1204.6456 . Google Scholar
H. Goda, T. Kitano and T. Morifuji, Comment. Math. Helv. 80, 51 (2005). Web of Science, Google Scholar
J. Hoste and P. Shanahan, J. Knot Theory Ramifications 22(1), 1250138 (2013), DOI: 10.1142/S0218216512501386. Link, Web of Science, Google Scholar
T. Kim and T. Morifuji, Internat. J. Math. 23, 1250022 (2012), DOI: 10.1142/S0129167X11007653. Link, Web of Science, Google Scholar
T. Kitano and T. Morifuji, Ann. Sci. Norm. Sup. Pisa Cl. Sci. (5) 4, 179 (2005). Google Scholar
T. Le, Varieties of representations and their subvarieties of cohomology jumps for knot groups, Mat. Sb. 184 (1993) 57–82 (in Russian); Russian Acad. Sci. Sb. Math. 78 (1994) 187–209 (in English) . Google Scholar
X. S. Lin, Acta Math. Sin. (English Ser.) 17, 361 (2001), DOI: 10.1007/s101140100122. Web of Science, Google Scholar
T. Morifuji and A. Tran, Twisted Alexander polynomials of 2-bridge knots for parabolic representations, preprint (2013) , arXiv:1301.1101 . Google Scholar
K. Murasugi, J. Math. Soc. Japan 10, 94 (1958), DOI: 10.2969/jmsj/01010094. Web of Science, Google Scholar
K. Murasugi, J. Math. Soc. Japan 10, 235 (1985), DOI: 10.2969/jmsj/01030235. Web of Science, Google Scholar
F. Nagasato and A. Tran, Presentations of character varieties of 2-bridge knots using Chebyshev polynomials, preprint (2013) , arXiv:1301.0138 . Google Scholar
A. Tran, On left-orderable fundamental groups and Dehn surgeries on twist knots, preprint (2013) , arXiv:1301.0665 . Google Scholar
M. Wada, Topology 33, 241 (1994), DOI: 10.1016/0040-9383(94)90013-2. Web of Science, Google Scholar