TY - JOUR
T1 - Twisted alexander polynomials and character varieties of 2-bridge knot groups
AU - Kim, Taehee
AU - Morifuji, Takayuki
N1 - Funding Information:
would like to thank Stefan Friedl for helpful comments and correcting the proof of Theorem 4.3. The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (Nos. 2009-0068877 and 2009-0086441). The second author was supported in part by the Grant-in-Aid for Scientific Research (No. 20740030), the Ministry of Education, Culture, Sports, Science and Technology, Japan.
PY - 2012/6
Y1 - 2012/6
N2 - We study the twisted Alexander polynomial from the viewpoint of the SL(2,&Cmathbb;)-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,&Cmathbb;)-representations are all monic. In this paper, we show that for a 2-bridge knot there exists a curve component in the SL(2,&Cmathbb;)-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.
AB - We study the twisted Alexander polynomial from the viewpoint of the SL(2,&Cmathbb;)-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,&Cmathbb;)-representations are all monic. In this paper, we show that for a 2-bridge knot there exists a curve component in the SL(2,&Cmathbb;)-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.
KW - 2-bridge knot
KW - Twisted Alexander polynomial
KW - character variety
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U2 - 10.1142/S0129167X11007653
DO - 10.1142/S0129167X11007653
M3 - Review article
AN - SCOPUS:84860508764
SN - 0129-167X
VL - 23
JO - International Journal of Mathematics
JF - International Journal of Mathematics
IS - 6
M1 - 1250022
ER -