Related papers: A table of genus two handlebody-knots with seven c…

A handlebody link is a union of handlebodies of positive genus embedded in 3-space, which generalizes the notion of links in classical knot theory. In this paper, we consider handlebody links with one genus 2 handlebody and $n-1$ solid…

We investigate the class of $3$-decomposable genus two handlebody-knots and provide a complete classification of essential annuli in their exteriors. We introduce the notion of $\tau$- and $\rho$-tangles and good rectangles and annuli. By…

We characterize composite tunnel number one genus two handlebody-knots.

We give examples of knots in a genus 2 handlebody which have nontrivial Dehn surgeries yielding handlebodies and show that these knots are not 1--bridge.

We describe the genus two knots which admit a genus one, one bridge position. These are divided into several families, one consists of vertical bandings of two genus one $(1,1)$-knots, other consists of vertical bandings of two cross cap…

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…

We present a practical algorithm to determine the minimal genus of non-orientable spanning surfaces for 2-bridge knots, called the crosscap numbers. We will exhibit a table of crosscap numbers of 2-bridge knots up to 12crossings (all 362 of…

We establish a Kauffman-Murasugi-Thistlethwaite-type theorem for alternating knots in a solid torus. Specifically, we show that any dotted-reduced alternating diagram of a knot in a handlebody realizes the minimal crossing number, and that…

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.

For any knot with genus one and unknotting number one, other than the figure-eight knot, we prove that there is exactly one way to unknot it by means of a crossing change. In the case of the figure-eight knot, we prove that there are…

For a knot $K$, the doubly slice genus $g_{ds}(K)$ is the minimal $g$ such that $K$ divides a closed, orientable, and unknotted surface of genus $g$ embedded in $S^4$. In this paper, we identify the doubly slice genera of 2909 of the 2977…

We show that for genus one knots the Alexander polynomial and the homology of the double cover branching over the knot provide obstructions to cosmetic crossings. As an application we prove the nugatory crossing conjecture for the…

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…

In this paper we define a set of numerical criteria for a handlebody link to be irreducible. It provides an effective, easy-to-implement method to determine the irreducibility of handlebody links; particularly, it recognizes the…

The paper applies the JSJ decomposition and Koda-Ozawa's annulus classification to analyze the annulus configuration in a handlebody-knot exterior. We introduce the notion of the annulus diagram, to pack the configuration into a labeled…

We present a complete classification of spherical knotoids with up to six crossings and conjecture that our classification up to seven crossings is complete. Our work extends the tradition of knot tabulation to the setting of knotoids…

The list of knots with up to 10 crossings is commonly referred to as the Rolfsen Table. This paper presents a way to generate the Rolfsen table in a simple, clear, and reproducible manner. The methods we use are similar to those used by J.…

We define a knot to be half ribbon if it is the cross-section of a ribbon 2-knot, and observe that ribbon implies half ribbon implies slice. We introduce the half ribbon genus of a knot K, the minimum genus of a ribbon knotted surface of…

We show that the triple-crossing number of any knot is greater or equal to twice its (canonical) genus and we show an even stronger bound in the case of links. As an application we show that this bound is strong enough to obtain the…

We give infinitely many examples of 2-bridge knots for which the topological and smooth slice genera differ. The smallest of these is the 12-crossing knot $12a255$. These also provide the first known examples of alternating knots for which…