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Register a new userTwisted Alexander polynomials and ideal points giving Seifert surfaces
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Authors
Takahiro Kitayama
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The coefficients of twisted Alexander polynomials of a knot induce regular functions of the $SL_2(mathbb{C})$-character variety. We prove that the function of the highest degree has a finite value at an ideal point which gives a minimal genus Seifert surface by Culler-Shalen theory. It implies a partial affirmative answer to a conjecture by Dunfield, Friedl and Jackson.
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We study incompressible surfaces constructed by Culler-Shalen theory in the context of twisted Alexander polynomials. For a $1$st cohomology class of a $3$-manifold the coefficients of twisted Alexander polynomials induce regular functions on the $SL_2(mathbb{C})$-character variety. We prove that if an ideal point gives a Thurston norm minimizing non-separating surface dual to the cohomology class, then the regular function of the highest degree has a finite value at the ideal point.
In this paper we give an explicit formula for the twisted Alexander polynomial of any torus link and show that it is a locally constant function on the $SL(2, mathbb C)$-character variety. We also discuss similar things for the higher dimensional twisted Alexander polynomial and the Reidemeister torsion.
In this paper we apply the twisted Alexander polynomial to study the fibering and genus detecting problems for oriented links. In particular we generalize a conjecture of Dunfield, Friedl and Jackson on the torsion polynomial of hyperbolic knots to hyperbolic links, and confirm it for an infinite family of hyperbolic 2-bridge links. Moreover we consider a similar problem for parabolic representations of 2-bridge link groups.
We establish homotopy ribbon concordance obstructions coming from the Blanchfield form and Levine-Tristram signatures. Then, as an application of twisted Alexander polynomials, we show that for every knot K with nontrivial Alexander polynomial, there exists an infinite family of knots that are all concordant to K and have the same Blanchfield form as K, such that no pair of knots in that family is homotopy ribbon concordant.
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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