Related papers: A note on Riley polynomials of $2$-bridge knots
Experimental data from Dunfield et al using random grid diagrams suggests that the genus of a knot grows linearly with respect to the crossing number. Using billiard table diagrams of Chebyshev knots developed by Koseleff and Pecker and a…
A polynomial is presented that models a topological knot in a unique manner. It distinguishes all types of knots including the orientation and has a group theory interpretation. The topologies may be labeled via a number, which upon a base…
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…
We give a complete characterization of those essential simple loops on 2-bridge spheres of 2-bridge links which are null-homotopic in the link complements. By using this result, we describe all upper-meridian-pair-preserving epimorphisms…
We describe the (P)SL(2,C) character varieties of all 2-bridge knots and the diagonal character varieties for all 2-bridge links in terms of a set of polynomials defined using Farey recursion.
We prove that for any zero {\alpha} of the Alexander polynomial of a two-bridge knot, -3 < Re({\alpha}) < 6. Furthermore, for a large class of two-bridge knots we prove -1<Re({\alpha}).
In this paper we use continued fractions to study a partial order on the set of 2-bridge knots derived from the work of Ohtsuki, Riley, and Sakuma. We establish necessary and sufficient conditions for any set of 2-bridge knots to have an…
In this paper, we discuss the region unknotting number of different classes of 2-bridge knots. In particular, we provide region unknotting number for the classes of $2$-bridge knots whose Conway notation is $C(m,\ n), C(m,\ 2,\ m),$ $ C(m,\…
We confirm the AJ conjecture [Ga04] that relates the A-polynomial and the colored Jones polynomial for those hyperbolic knots satisfying certain conditions. In particular, we show that the conjecture holds true for some classes of…
We prove that all 2-bridge ribbon knots are symmetric unions.
We generalize R. Riley's study about parabolic representations of two bridge knot groups to the general knots in $S^3$. We utilize the parabolic quandle method for general knot diagrams and adopt symplectic quandle for better investigation,…
We show that every two-bridge knot $K$ of crossing number $N$ admits a polynomial parametrization $x=T_3(t), y = T_b(t), z =C(t)$ where $T_k(t)$ are the Chebyshev polynomials and $b+\deg C = 3N$. If $C (t)= T_c(t)$ is a Chebyshev…
Symmetries of knots have been studied extensively, and strongly invertible knots are one of them. Lamm defined the equivariant crossing number $c_t(K)$, the minimum crossing number among all symmetric diagrams for a strongly invertible knot…
We publish a table of primitive finite-type invariants of order less than or equal to six, for knots of ten or fewer crossings. We note certain mod-2 congruences, one of which leads to a chirality criterion in the Alexander polynomial. We…
We use twisted Alexander polynomials to show that certain algebraically slice 2-bridge knots are not topologically slice, even though all prime power Casson-Gordon signatures vanish. We also provide some computations indicating the efficacy…
We give an alternative proof of a result of Kobayashi and Saeki that every genus one $1$-bridge position of a non-trivial $2$-bridge knot is a stabilization.
We compute Cayley graphs and automorphism groups for all finite $n$-quandles of two-bridge and torus knots and links, as well as torus links with an axis.
We investigate the twisted Alexander polynomial of a 2-bridge knot associated to a Fox coloring. For several families of 2-bridge knots, including but not limited to, torus knots and genus-one knots, we derive formulae for these twisted…
We prove that hyperbolic 2-bridge knots are determined amongst all compact 3-manifolds by the profinite completions of their knot groups.
Residual torsion-free nilpotence has proven to be an important property for knot groups with applications to bi-orderability and ribbon concordance. Mayland proposed a strategy to show that a two-bridge knot group has a commutator subgroup…