Narrow Results

We give explicit formulas for the adjoint twisted Alexander polynomial and nonabelian Reidemeister torsion of genus one two-bridge knots.

We give explicit formulas 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.

We consider cyclic branched coverings of a 3-parameter family of rational knots in $S^3$ and study the left orderability of their fundamental groups. We first compute the nonabelian ${\mathrm{\text{SL}}}_2(ℂ)$-character varieties of the rational knots $C(2p,2m,2n+1)$ in the Conway notation, where $p,m,n$ are integers. We then study real points on these varieties and finally use them to determine the left orderability of the fundamental groups of cyclic branched coverings of $C(2p,2m,2n+1)$.

A nontrivial knot is called minimal if its knot group does not surject onto the knot groups of other nontrivial knots. In this paper, we determine the minimality of the rational knots $C(2n+1,2m,2)$ in the Conway notation, where $m\ne 0$ and $n\ne 0,-1$ are integers. When $\left|m\right|\ge 2$, we show that the nonabelian ${\mathrm{\text{SL}}}_2(ℂ)$-character variety of $C(2n+1,2m,2)$ is irreducible and therefore $C(2n+1,2m,2)$ is a minimal knot. The proof of this result is an interesting application of Eisenstein’s irreducibility criterion for polynomials over integral domains.