Related papers: Torus knots obtained by negatively twisting torus …

We obtain the full list of Goeritz invariants of all torus knots and links.

In this paper we study welded knots and their invariants. We focus on generating examples of non-trivial knotted ribbon tori as the tube of welded knots that are obtained from classical knot diagrams by welding some of the crossings.…

The twisted torus link $T(p, q; r, s)$ is obtained by twisting $r$ parallel strands of the $(p, q)$-torus link a total of $s$ full times. In this paper we find all twisted torus links which are hyperbolic for $\vert s\vert >3$.

We study simple, knotted and linked torus windings that are made of tubes of finite thickness. Knots which have the shortest rope length are often denoted ideal structures. Conventionally, the ideal structure are found by rope shortening…

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 give a complete coarse classification of Legendrian and transverse torus knots in any contact structure on $S^3$.

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…

We show that for many classical knots one can find generalized torsion in the fundamental group of its complement, commonly called the knot group. It follows that such a group is not bi-orderable. Examples include all torus knots, the…

The curves of zero intensity of a complex optical field can form knots and links: optical vortex knots. Both theoretical constructions and experiments have so far been restricted to the very small families of torus knots or lemniscate…

Knot contact homology is an invariant of knots derived from Legendrian contact homology which has numerous connections to the knot group. We use basic properties of knot groups to prove that knot contact homology detects every torus knot.…

We give some conditions on positive braids with at least two full twists that ensure their closure is a hyperbolic knot, with applications to the geometric classification of T-links, arising from dynamics, and twisted torus knots.

The ribbonlength Rib$(K)$ of a knot $K$ is the infimum of the ratio of the length of any flat knotted ribbon with core $K$ to its width. A twisted torus knot $T_{p,q;r,s}$ is obtained from the torus knot $T_{p,q}$ by twisting $r$ adjacent…

In the search for transverse-universal knots in the standard contact structure on $\mathbb{S}^3$, we present a classification of the transverse twist knots with maximal self-linking number, that admit only overtwisted contact branched…

The crosscap number of a knot in the 3-sphere is the minimal genus of non-orientable surface bounded by the knot. We determine the crosscap numbers of torus knots.

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…

We determine the locally flat cobordism distance between torus knots with small and large braid index, up to high precision. Here small means 2, 3, 4, or 6. As an application, we derive a surprising fact about torus knots that appear as…

In a group, a non-trivial element is called a generalized torsion element if some non-empty finite product of its conjugates equals to the identity. We say that a knot has generalized torsion if its knot group admits such an element. For a…

In math.GT/0002110 the author's Theorems 1.1 and 1.2, combined, implied that iterated torus knots are transversally simple. This result is in error and this erratum pin points the error. In "An addendum on iterated torus knots" a more…

Let $r$ be an odd integer, $r\ge3$. Then the petal number of the torus knot of type $(r,r+2)$ is equal to $2r+3$.

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…