TY - JOUR
T1 - Twisted Alexander polynomials, character varieties and Reidemeister torsions of double branched covers
AU - Yamaguchi, Yoshikazu
N1 - Funding Information:
The author wishes to express his thanks to Professor Kunio Murasugi for drawing the author's attention to the classification of two-bridge knots by the twisted Alexander polynomial. He also would like to thank the referee for his/her careful reading and helpful comments to improve the manuscript. This research was supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists and Grant-in-Aid for JSPS Fellows Grant Number 10J07961 .
Publisher Copyright:
© 2016 Elsevier B.V.
PY - 2016/5/15
Y1 - 2016/5/15
N2 - We give an extension of Fox's formula of the Alexander polynomial for a double branched cover over the three-sphere. Our formula provides the Reidemeister torsion of the double branched cover along a knot for a non-trivial 1-dimensional representation. In our formula, the Reidemeister torsion is given by the product of two factors derived from the knot group. One of the factors is determined by the twisted Alexander polynomial and the other is determined by a rational function on the character variety of the knot group. As an application, we show that these products distinguish the isotopy classes of two-bridge knots up to their mirror images.
AB - We give an extension of Fox's formula of the Alexander polynomial for a double branched cover over the three-sphere. Our formula provides the Reidemeister torsion of the double branched cover along a knot for a non-trivial 1-dimensional representation. In our formula, the Reidemeister torsion is given by the product of two factors derived from the knot group. One of the factors is determined by the twisted Alexander polynomial and the other is determined by a rational function on the character variety of the knot group. As an application, we show that these products distinguish the isotopy classes of two-bridge knots up to their mirror images.
KW - Branched coverings
KW - Character varieties
KW - Knots
KW - Metabelian representations
KW - Primary
KW - Reidemeister torsion
KW - Secondary
KW - Twisted Alexander polynomial
UR - http://www.scopus.com/inward/record.url?scp=84961894512&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84961894512&partnerID=8YFLogxK
U2 - 10.1016/j.topol.2016.03.021
DO - 10.1016/j.topol.2016.03.021
M3 - Article
AN - SCOPUS:84961894512
SN - 0166-8641
VL - 204
SP - 278
EP - 305
JO - Topology and its Applications
JF - Topology and its Applications
ER -