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We give a new interpretation of the Alexander polynomial $Delta_0$ for virtual knots due to Sawollek and Silver and Williams, and use it to show that, for any virtual knot, $Delta_0$ determines the writhe polynomial of Cheng and Gao (equivalently, Kauffmans affine index polynomial). We also use it to define a second-order writhe polynomial, and give some applications.
The writhe polynomial is a fundamental invariant of an oriented virtual knot. We introduce a kind of local moves for oriented virtual knots called shell moves. The first aim of this paper is to prove that two oriented virtual knots have the same writ
We define a family of virtual knots generalizing the classical twist knots. We develop a recursive formula for the Alexander polynomial $Delta_0$ (as defined by Silver and Williams) of these virtual twist knots. These results are applied to provide e
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.
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
Study of certain isotopy classes of a finite collection of immersed circles without triple or higher intersections on closed oriented surfaces can be thought of as a planar analogue of virtual knot theory where the genus zero case corresponds to clas