Related papers: Additional cases of positive twisted torus knots

A Seifert surgery is an integral surgery on a knot in S^3 producing a Seifert fiber space which may contain an exceptional fiber of index 0. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert…

A Lissajous knot is one that can be parameterized by a single cosine function in each coordinate. Lissajous knots are highly symmetric, and for this reason, not all knots are Lissajous. We prove several theorems which allow us to place…

A 1-bridge torus knot in a 3-manifold of genus $\le 1$ is a knot drawn on a Heegaard torus with one bridge. We give two types of normal forms to parameterize the family of 1-bridge torus knots that are similar to the Schubert's normal form…

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…

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.…

A knot K is called Gordian adjacent to a knot L if there exists an unknotting sequence for L containing K. We provide a sufficient condition for Gordian adjacency of torus knots via the study of knots in the thickened torus. We also…

All knots are known to be hyperbolic, satellite, or torus knots, and one important family is Lorenz links, or T-links, which arise from dynamics. However, it remains difficult to determine the geometric type of a Lorenz link from a…

The nonorientable four-ball genus of a knot $K$ in $S^3$ is the minimal first Betti number of nonorientable surfaces in $B^4$ bounded by $K$. By amalgamating ideas from involutive knot Floer homology and unoriented knot Floer homology, we…

The A-polynomial of a knot is defined in terms of SL(2,C) representations of the knot group, and encodes information about essential surfaces in the knot complement. In 2005, Dunfield-Garoufalidis and Boyer-Zhang proved that it detects the…

In Theorem 1.2 of the paper math.GT/0002110 the author claimed to have proved that all transversal knots whose topological knot type is that of an iterated torus knot (we call them cable knots) are transversally simple. That theorem is…

We give a complete classification of toroidal Seifert fibered surgeries on alternating knots. Precisely, we show that if an alternating knot admits a toroidal Seifert fibered surgery, then the knot is either the trefoil knot and the surgery…

It is a well-known procedure for constructing a torus knot or link that first we prepare an unknotted torus and meridian disks in the complementary solid tori of it, and second smooth the intersections of the boundary of meridian disks…

We show that every non-trivial tame knot or link in R^3 has a quadrisecant, i.e. four collinear points. The quadrisecant must be topologically non-trivial in a precise sense. As an application, we show that a nonsingular, algebraic surface…

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

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…

Let K \subset Y be a knot in a three manifold which admits a longitude-framed surgery such that the surgered manifold has first Betti number greater than that of Y. We find a formula which computes the twisted Floer homology of the surgered…

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…

A knot type is exchange reducible if an arbitrary closed n-braid representative can be changed to a closed braid of minimum braid index by a finite sequence of braid isotopies, exchange moves and +/- destabilizations. In the manuscript [J…

All knots in $R^3$ possess Seifert surfaces, and so the classical Thurston-Bennequin and rotation (or Maslov) invariants for Legendrian knots in a contact structure on $R^3$ can be defined. The definitions extend easily to null-homologous…

A torus-covering $T^2$-knot is a surface-knot of genus one determined from a pair of commutative braids. For a torus-covering $T^2$-knot $F$, we determine the number of irreducible metabelian $SU(2)$-representations of the knot group of $F$…