Related papers: Epimorphisms between $2$-bridge knot groups and th…

We have the generating function which determines the number of $2$-bridge knot groups admitting epimorphisms onto the knot group of a given $2$-bridge knot, in terms of crossing number. In this paper, we will refine this formula by taking…

Let $K,K'$ be two-bridge knots of genus $n,k$ respectively. We show the necessary and sufficient condition of $n$ in terms of $k$ that there exists an epimorphism from the knot group of $K$ onto that of $K'$.

We consider the relations $\ge$ and $\ge_p$ on the collection of all knots, where $k \ge k'$ (respectively, $k \ge_p k'$) if there exists an epimorphism $\pi k \to \pi k'$ of knot groups (respectively, preserving peripheral systems). When…

We consider the relationship between the crosscap number $\gamma$ of knots and a partial order on the set of all prime knots, which is defined as follows. For two knots $K$ and $J$, we say $K \geq J$ if there exists an epimorphism…

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…

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…

We show that any parabolic generating pair of a genus-one hyperbolic 2-bridge knot group is equivalent to the upper or lower meridian pair. As an application, we obtain a complete classification of the epimorphisms from 2-bridge knot groups…

In this paper, we determine the average genus of all the $2$-bridge knots with a given crossing number. As a consequence, we obtain the oblique asymptote of this value as the crossing number grows.

In this short note we show the existence of an epimorphism between groups of $2$-bridge knots by means of an elementary argument using the Riley polynomial. As a corollary, we give a classification of $2$-bridge knots by Riley polynomials.

Negami found an upper bound on the stick number $s(K)$ of a nontrivial knot $K$ in terms of the minimal crossing number $c(K)$ of the knot which is $s(K) \leq 2 c(K)$. Furthermore McCabe proved $s(K) \leq c(K) + 3$ for a $2$-bridge knot or…

In this article we study a partial ordering on knots in the 3-sphere where K_1 is greater than or equal to K_2 if there is an epimorphism from the knot group of K_1 onto the knot group of K_2 which preserves peripheral structure. If K_1 is…

Riley "defined" the Heckoid groups for 2-bridge links as Kleinian groups, with nontrivial torsion, generated by two parabolic transformations, and he constructed an infinite family of epimorphisms from 2-bridge link groups onto Heckoid…

For any given number of crossings $c$, there exists a formula to determine the number of 2-bridge knots of $c$ crossings, and indeed it is a simple matter to actually construct presentations of these knots. However, the determination of…

Experimental work suggests that the Seifert genus of a knot grows linearly with respect to the crossing number of the knot. In this article, we use a billiard table model for $2$-bridge or rational knots to show that the average genus of a…

In Part I of this series of papers, we made Riley's definition of Heckoid groups for 2-bridge links explicit, and gave a systematic construction of epimorphisms from 2-bridge link groups onto Heckoid groups, generalizing Riley's…

A partial order on prime knots can be defined by declaring $J\ge K$ if there exists an epimorphism from the knot group of $J$ onto the knot group of $K$. Suppose that $J$ is a 2-bridge knot that is strictly greater than $m$ distinct,…

We give an alternative proof to Agol's classification of parabolic generating pairs of non-free Kleinian groups generated by two parabolic transformations. As an application, we give a complete characterisation of epimorphims between…

The average genus of a 2-bridge knot with crossing number $c$ approaches $\frac{c}{4} + \frac{1}{12}$ as $c$ approaches infinity, as proven by Suzuki and Tran and independently Cohen and Lowrance. In this paper, for the genera of $2$-bridge…

We give a systematic construction of epimorphisms between 2-bridge link groups. Moreover, we show that 2-bridge links having such an epimorphism between their link groups are related by a map between the ambient spaces which only have a…

We investigate the bi-orderability of two-bridge knot groups and the groups of knots with 12 or fewer crossings by applying recent theorems of Chiswell, Glass and Wilson. Amongst all knots with 12 or fewer crossings (of which there are…