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Register a new userEvaluations of the twisted Alexander polynomials of 2-bridge knots at $pm 1$
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Authors
Mikami Hirasawa
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Let $H(p)$ be the set of 2-bridge knots $K$ whose group $G$ is mapped onto a non-trivial free product, $Z/2 * Z/p$, $p$ being odd. Then there is an algebraic integer $s_0$ such that for any $K$ in $H(p)$, $G$ has a parabolic representation $rho$ into $SL(2, Z[s_0]) subset SL(2,C)$. Let $Delta(t)$ be the twisted Alexander polynomial associated to $rho$. Then we prove that for any $K$ in $H(p)$, $Delta(1)=-2s_0^{-1}$ and $Delta(-1)=-2s_0^{-1}mu^2$, where $s_0^{-1}, mu in Z[s_0]$. The number $mu$ can be recursively evaluated.
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In this paper we show that the twisted Alexander polynomial associated to a parabolic representation determines fiberedness and genus of a wide class of 2-bridge knots. As a corollary we give an affirmative answer to a conjecture of Dunfield, Friedl and Jackson for infinitely many hyperbolic knots.
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 the converse holds for 2-bridge knots. Furthermore 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.
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