Narrow Results

We prove a simple necessary and sufficient condition for a two-bridge knot $K(p,q)$ to be quasipositive, based on the continued fraction expansion of $p/q$. As an application, coupled with some classification results in contact and symplectic topology, we give a new proof of the fact that smoothly slice two-bridge knots are non-quasipositive. Another proof of this fact using methods within the scope of knot theory is presented in Appendix A, by Stepan Orevkov.

We show that there is no non-trivial 3-adjacency relation between two-bridge knots or links. As an application of our proof argument, we also show that if a knot K is 3-adjacent to a two-bridge knot or fibered knot K′, then K and K′ have the same determinant.

A conjecture of Riley about the relationship between real parabolic representations and signatures of two-bridge knots is verified for double twist knots.

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.

In this paper, we study the Riley polynomial of double twist knots with higher genus. Using the root of the Riley polynomial, we compute the range of rational slope $r$ such that $r$-filling of the knot complement has left-orderable fundamental group. Further more, we make a conjecture about left-orderable surgery slopes of two-bridge knots.

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.

We show that a two-bridge ribbon knot $K(m^2,mk\pm 1)$ with $m>k>0$ and $(m,k)=1$ admits a symmetric union presentation with partial knot which is a two-bridge knot $K(m,k)$. Similar descriptions for all the other two-bridge ribbon knots are also given.

In this paper, we consider two properties on the braid index of a two-bridge knot. We prove an inequality on the braid indices of two-bridge knots if there exists an epimorphism between their knot groups. Moreover, we provide the average braid index of all two-bridge knots with a given crossing number.