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Register a new userTwisted Alexander polynomials of torus links
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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.
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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.
This paper is a continuation on the 2012 paper on Cutting Twisted Solid Tori (TSTs), in which we considered twisted solid torus links (tst links). We generalize the notion of tst links to surgerized tst links: recall that when performing $Phi^mu(n(tau), d(tau), M)$ on a tst $langle tau rangle$ where $M$ is odd, we obtain the tst link, $[Phi^mu(n(tau), d(tau), M)]$ that contains a trivial knot as one of its components. We then perform another operation $Phi^{mu}(n(tau ), d(tau ), M)$ on that trivial knot to create a new link, which we call a surgerized tst link (stst link). If $M$ is odd, we can repeat the process to give more complicated stst links. We compute braid words, Alexander and Jones polynomials of such links.
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.
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.
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.
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