Related papers: Crossing numbers of cable knots

This paper investigates the relationship between the signature and the crossing number of knots and links. We refine existing theorems and provide a comprehensive classification of links with specific properties, particularly those with…

The paper introduces Slope Conjecture which relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. More precisely, we introduce two knot invariants, the…

We show that most cabled knots over torus knots in $S^3$ satisfy the AJ-conjecture, namely each $(r,s)$-cabled knot over each $(p,q)$-torus knot satisfies the $AJ$-conjecture if $r$ is not a number between $0$ and $pqs$.

Levine defined the rational algebraic knot concordance group and proved that each nontrivial element is of order two, of order four, or of infinite order. The determination of the order of an element depends on a p-adic analysis for all…

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 give examples of knots with some unusual properties of the crossing number of positive diagrams or strand number of positive braid representations. In particular we show that positive braid knots may not have positive minimal (strand…

We show that a finite numerical boundary slope of an essential surface in the exterior of a Montesinos knot is bounded above and below in terms of the numbers of positive/negative crossings of a specific minimal diagram of the knot.

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…

The twisting number of a ribbon knot $K$ is the minimal number of tangle replacements on the symmetry axis of $J \# -J$ for any knot $J$ that is required to produce a symmetric union diagram of $K$. We prove that the twisting number is…

We present two families of knots which have straight number higher than crossing number. In the case of the second family, we have computed the straight number explicitly. We also give a general theorem about alternating knots that states…

We show that the head and tail functions of the colored Jones polynomial of adequate links are the product of head and tail functions of the colored Jones polynomial of alternating links that can be read-off an adequate diagram of the link.…

We use the rational Witt class of a knot in the 3-sphere as a tool for addressing questions about its unknotting number. We apply these tools to several low crossing knots (151 knots with 11 crossing and 100 knots with 12 crossings) and to…

Knot mosaics are used to model physical quantum states. The mosaic number of a knot is the smallest integer $m$ such that the knot can be represented as a knot $m$-mosaic. In this paper we establish an upper bound for the crossing number of…

We prove that for 2-bridge knots, the diameter, D, of the set of boundary slopes is twice the crossing number, c. This constitutes partial verification of a conjecture that, for all knots in S^3, D is at most 2c.

Suppose that there exists an epimorphism from the knot group of a $2$-bridge knot $K$ onto that of another knot $K'$. In this paper, we study the relationship between their crossing numbers $c(K)$ and $c(K')$. Especially it is shown that…

We show for an alternating knot the minimal boundary slope of an essential spanning surface is given by the signature plus twice the minimum degree of the Jones polynomial and the maximal boundary slope of an essential spanning surface is…

The crossing number of a graph is the minimum number of edge crossings that a graph can have when drawn in the plane. Determining this number, known as the Crossing Number problem, is a celebrated problem in combinatorial optimization. It…

We show examples of knots with the same polynomial invariants and hyperbolic volumes, with variously coinciding 2-cable polynomials and colored Jones polynomials, which are not mutants.

We compute the bridge spectra of cables of 2-bridge knots. We also give some results about bridge spectra and distance of Montesinos knots.

We show that a knot whose minimum crossing number $c(K)$ is even and greater than $30$ is not fertile; there exists a knot $K'$ with crossing number less than $c$ such that $K'$ is not obtained from a minimum crossing number diagram of $K$…