Related papers: The unknotting number and classical invariants I

We investigate several integer invariants of curves in 3-space. We demonstrate relationships of these invariants to crossing number and to total curvature.

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…

We investigate the behaviour of Rasmussen's invariant $s$ under the sharp operation on knots and obtain a lower bound for the sharp unknotting number. This bound leads us to an interesting move that transforms arbitrary knots into…

We use a version of simulated annealing with knot-type preserving moves to find polygonal representatives of various knot types with low stick number. These give better bounds on stick numbers of prime knots through 10 crossings, and for…

The ribbon number of a knot is the minimum number of ribbon singularities among all ribbon disks bounded by that knot. In this paper, we build on the systematic treatment of this knot invariant initiated in recent work of Friedl, Misev, and…

The concordance genus of a knot is the least genus of any knot in its concordance class. It is bounded above by the genus of the knot, and bounded below by the slice genus, two well-studied invariants. In this paper we consider the…

A triple crossing is a crossing in a projection of a knot or link that has three strands of the knot passing straight through it. A triple crossing projection is a projection such that all of the crossings are triple crossings. We prove…

We introduce a framework to analyze knots and links in an unmarked solid torus. We discuss invariants that detect when such links are equivalent under an ambient homeomorphism, and show that the multivariable Alexander polynomial is such in…

We prove that the degree of the Brandt-Lickorish-Millet polynomial of any quasi-alternating link is less than its determinant. Therefore, we obtain a new and a simple obstruction criterion for quasi-alternateness. As an application, we…

The knot Floer order $\operatorname{Ord}(K)$ is a knot invariant derived from knot Floer homology that provides bounds on many other invariants, such as the bridge index $\operatorname{br}(K)$ for which $\operatorname{Ord}(K) + 1 \leq…

Chern-Simons gauge theory for compact semisimple groups is analyzed from a perturbation theory point of view. The general form of the perturbative series expansion of a Wilson line is presented in terms of the Casimir operators of the gauge…

Using region crossing changes, we define a new invariant called the multi-region index of a knot. We prove that the multi-region index of a knot is bounded from above by twice the crossing number of the knot. In addition, we show that the…

Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by V.~Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter…

A knot is an an embedding of a circle into three-dimensional space. We say that a knot is unknotted if there is an ambient isotopy of the embedding to a standard circle. By representing knots via planar diagrams, we discuss the problem of…

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…

Using the knot Floer homology filtration, we define invariants associated to a knot in a three-manifold possessing non-vanishing Floer co(homology) classes. In the case of the Ozsvath-Szabo contact invariant we obtain an invariant of knots…

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…

We prove that fibred knots cannot be untied with $\bar{t}_{2k}$-moves, for all $k \geq 2$. More generally, we give an upper bound on the number of two strand twist operations that allow to untie a knot with non-trivial HOMFLY polynomial, in…

Traditionally, knot theorists have considered projections of knots where there are two strands meeting at every crossing. A triple crossing is a crossing where three strands meet at a single point, such that each strand bisects the…

In this paper, we show the trivializing number of all minimal diagrams of positive 2-bridge knots and study the relation between the trivializing number and the unknotting number for a part of these knots.