Related papers: Unknotting genus one knots

We classify all knot diagrams of genus two and three, and give applications to positive, alternating and homogeneous knots, including a classification of achiral genus 2 alternating knots, slice or achiral 2-almost positive knots, a proof…

A knot in the 3-sphere is said to have zero negative unknotting number if it can be transformed into the unknot by performing only positive crossing changes. In this paper, we provide an obstruction for a knot to having zero negative…

We show that there are hyperbolic tunnel-number one knots with arbitrarily high bridge number and that "most" tunnel-number one knots are not one-bridge with respect to an unknotted torus. The proof relies on a connection between bridge…

Given a diagram $D$ of a knot $K$, we consider the number $c(D)$ of crossings and the number $b(D)$ of overpasses of $D$. We show that, if $D$ is a diagram of a nontrivial knot $K$ whose number $c(D)$ of crossings is minimal, then…

We compose the table of knots in the thickened torus T x I having diagrams with at most 4 crossings. The knots are constructed by the three-step process. First we list regular graphs of degree 4 with at most 4 vertices, then for each graph…

Whitehead doubles provide a plethora of examples of knots that are topologically slice but not smoothly slice. We discuss the problem of the Whitehead double of the Figure 8 knot and survey commonly used techniques to obstructing sliceness.…

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 explore the application of automated reasoning techniques to unknot detection, a classical problem of computational topology. We adopt a two-pronged experimental approach, using a theorem prover to try to establish a positive result…

The only knots that are tunnel number one and genus one are those that are already known: 2-bridge knots obtained by plumbing together two unknotted annuli and the satellite examples classified by Eudave-Munoz and by Morimoto-Sakuma. This…

In this paper, we study the unknotting operation for twisted knots, called arc shift move. First, we find a family of twisted knots with arc shift number $n$ for any given $n \in \mathbb{N}$. Then we define a new unknotting operation,…

For a genus-1 1-bridge knot in the 3-sphere, that is, a (1,1)-knot, a middle tunnel is a tunnel that is not an upper or lower tunnel for some (1,1)-position. Most torus knots have a middle tunnel, and non-torus-knot examples were obtained…

Kearton observed that mutation can change the concordance class of a knot. A close examination of his example reveals that it is of 4-genus 1 and has a mutant of 4-genus 0. The first goal of this paper is to construct examples to show that…

In this note we use Blanchfield forms to study knots that can be turned into an unknot using a single $\overline{t}_{2k}$ move.

Numerical simulations indicate that there exist conformations of the unknot, tied on a finite piece of rope, entangled in such a manner, that they cannot be disentangled to the torus conformation without cutting the rope. The simplest…

An open question asks if every knot of 4-genus g_s can be changed into a slice knot by g_s crossing changes. A counterexample is given.

We present two different representations of (1,1)-knots and study some connections between them. The first representation is algebraic: every (1,1)-knot is represented by an element of the pure mapping class group of the twice punctured…

For a knot diagram we introduce an operation which does not increase the genus of the diagram and does not change its representing knot type. We also describe a condition for this operation to certainly decrease the genus. The proof…

For $p\geq 1$ one can define a generalization of the unknotting number $tu_p$ called the $p$th untwisting number which counts the number of null-homologous twists on at most $2p$ strands required to convert the knot to the unknot. We show…

Generalizing unknotting number, $n$-adjacent knots have $n$ crossings such that changing any non-empty subset of them results in the unknot. In this paper, we determine the 2-adjacent knots through 12 crossings. Using Heegaard Floer…

We present the complete classification of the subgroup of the classical knot concordance group generated by knots with eight or fewer crossings. Proofs are presented in summary. We also describe extensions of this work to the case of nine…