The universal character ring of the (-2,2m+1,2n)-pretzel link

We explicitly calculate the universal character ring of the (-2,2m+1,2n)-pretzel link and show that it is reduced for all integers m and n.Comment: Very minor changes. To appear in the International Journal of Mathematic

On the AJ conjecture for cables of twist knots

We study the AJ conjecture that relates the A-polynomial and the colored Jones polynomial of a knot in $S^3$. We confirm the AJ conjecture for $(r,2)$-cables of the $m$-twist knot, for all odd integers $r$ satisfying $\begin{cases} (r+8)(r-8m)>0 &{if~} m> 0, \\ r(r+8m-4)>0 &{if~} m<0.\end{cases}$Comment: Results in Section 3 are corrected. arXiv admin note: text overlap with arXiv:1111.5258, arXiv:1405.4055; and with arXiv:math/0407521 by other author

On left-orderable fundamental groups and Dehn surgeries on knots

We show that the resulting manifold by $r$-surgery on a large class of two-bridge knots has left-orderable fundamental group if the slope $r$ satisfies certain conditions. This result gives a supporting evidence to a conjecture of Boyer, Gordon and Watson that relates $L$-spaces and the left-orderability of their fundamental groups.Comment: Very minor changes. To appear in the Journal of the Mathematical Society of Japa

The strong AJ conjecture for cables of torus knots

The AJ conjecture, formulated by Garoufalidis, relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. It has been confirmed for all torus knots, some classes of two-bridge knots and pretzel knots, and most cabled knots over torus knots. The strong AJ conjecture, formulated by Sikora, relates the A-ideal and the colored Jones polynomial of a knot. It was confirmed for all torus knots. In this paper we confirm the strong AJ conjecture for most cabled knots over torus knots.Comment: 9 page

Volumes of double twist knot cone-manifolds

We give explicit formulae for the volumes of hyperbolic cone-manifolds of double twist knots, a class of two-bridge knots which includes twist knots and two-bridge knots with Conway notation $C(2n,3)$. We also study the Riley polynomial of a class of one-relator groups which includes two-bridge knot groups.Comment: 11 pages, 1 figur

On left-orderability and cyclic branched coverings

In a recent paper Y. Hu has given a sufficient condition for the fundamental group of the r-th cyclic branched covering of S^3 along a prime knot to be left-orderable in terms of representations of the knot group. Applying her criterion to a large class of two-bridge knots, we determine a range of the integer r>1 for which the r-th cyclic branched covering of S^3 along the knot is left-orderable.Comment: 9 pages, 1 figur

Twisted Alexander polynomials with the adjoint action for some classes of knots

We calculate the twisted Alexander polynomial with the adjoint action for torus knots and twist knots. As consequences of these calculations, we obtain the formula for the nonabelian Reidemeister torsion of torus knots in \cite{Du} and a formula for the nonabelian Reidemeister torsion of twist knots that is better than the one in \cite{DHY}.Comment: 9 pages, 1 figure. To appear in Journal of Knot Theory and Its Ramification

On the non-existence of limit E-Brody curves

We give a simple proof of the non-existence of limit E-Brody curves, in the sense of Do Duc Thai, Mai Anh Duc and Ninh Van Thu, for a class of manifolds including $\mathbb{C}^n$ and $(\mathbb{C}^{\ast})^2$ which were studied by these authors in \cite{Do DT et al}, by constructing a suitable holomorphic interpolation function.Comment: 3 pages, some typos are corrected; Accepted for publication in Acta Mathematica Vietnamic

Nonabelian representations and signatures of double twist knots

A conjecture of Riley about the relationship between real parabolic representations and signatures of two-bridge knots is verified for double twist knots.Comment: 7 pages, 1 figur

Reidemeister torsion and Dehn surgery on twist knots

We compute the Reidemeister torsion of the complement of a twist knot in $S^3$ and that of the 3-manifold obtained by a Dehn surgery on a twist knot.Comment: 8 pages, 1 figur