On SL(3,C\mathbb C)-representations of the Whitehead link group

We describe a family of representations in SL(3,$\mathbb C$) of the fundamental group $\pi$ of the Whitehead link complement. These representations are obtained by considering pairs of regular order three elements in SL(3,$\mathbb C$) and can be seen as factorising through a quotient of $\pi$ defined by a certain exceptional Dehn surgery on the Whitehead link. Our main result is that these representations form an algebraic component of the SL(3,$\mathbb C$)-character variety of $\pi$.Comment: 20 pages, 3 figures, 4 tables, and a companion Sage notebook (see the references) v2: A few corrections and improvement

The SL(2,C) Casson invariant for Dehn surgeries on two-bridge knots

We investigate the behavior of the SL(2,C) Casson invariant for 3-manifolds obtained by Dehn surgery along two-bridge knots. Using the results of Hatcher and Thurston, and also results of Ohtsuki, we outline how to compute the Culler--Shalen seminorms, and we illustrate this approach by providing explicit computations for double twist knots. We then apply the surgery formula of Curtis to deduce the SL(2,C) Casson invariant for the 3-manifolds obtained by p/q-Dehn surgery on such knots. These results are applied to prove nontriviality of the SL(2,C) Casson invariant for nearly all 3-manifolds obtained by nontrivial Dehn surgery on a hyperbolic two-bridge knot. We relate the formulas derived to degrees of A-polynomials and use this information to identify factors of higher multiplicity in the $\hat{A}$-polynomial, which is the A-polynomial with multiplicities as defined by Boyer-Zhang.Comment: 32 pages, 2 figures, to be published in Algebraic and Geometric Topolog

Twisted Alexander polynomials and character varieties of 2-bridge knot groups

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.Comment: 19 pages, 1 figure, revised versio

Dimension of character varieties for 33-manifolds

Let $M$ be a $3$-manifold, compact with boundary and $\Gamma$ its fundamental group. Consider a complex reductive algebraic group G. The character variety $X(\Gamma,G)$ is the GIT quotient $\mathrm{Hom}(\Gamma,G)//G$ of the space of morphisms $\Gamma\to G$ by the natural action by conjugation of $G$. In the case $G=\mathrm{SL}(2,\mathbb C)$ this space has been thoroughly studied. Following work of Thurston, as presented by Culler-Shalen, we give a lower bound for the dimension of irreducible components of $X(\Gamma,G)$ in terms of the Euler characteristic $\chi(M)$ of $M$, the number $t$ of torus boundary components of $M$, the dimension $d$ and the rank $r$ of $G$. Indeed, under mild assumptions on an irreducible component $X_0$ of $X(\Gamma,G)$, we prove the inequality $$\mathrm{dim}(X_0)\geq t \cdot r - d\chi(M).$$Comment: 12 pages, 1 figur