Narrow Results

A twisted torus knot $T(p,q,r,s)$ is obtained from a torus knot $T(p,q)$ by twisting $r$ adjacent strands of $T(p,q)$ fully $s$ times. In this paper, we determine the parameters $p,q,r,s$ for which $T(p,q,r,s)$ is a torus knot with $s>0$.

A comprehensive study of geometric and topological properties of torus knots and unknots is presented. Torus knots/unknots are particularly symmetric, closed, space curves, that wrap the surface of a mathematical torus a number of times in the longitudinal and meridian direction. By using a standard parametrization, new results on local and global properties are found. In particular, we demonstrate the existence of inflection points for a given critical aspect ratio, determine the location and prescribe the regularization condition to remove the local singularity associated with torsion. Since to first approximation total length grows linearly with the number of coils, its nondimensional counterpart is proportional to the topological crossing number of the knot type. We analyze several global geometric quantities, such as total curvature, writhing number, total torsion, and geometric ‘energies’ given by total squared curvature and torsion, in relation to knot complexity measured by the winding number. We conclude with a brief presentation of research topics, where geometric and topological information on torus knots/unknots finds useful application.

It has long been conjectured that the crossing numbers of links are additive under the connected sum of links. This is a difficult problem in knot theory and has been open for more than 100 years. In fact, many questions of much weaker conditions are still open. For instance, it is not known whether Cr(K₁#K₂)≥Cr(K₁) or Cr(K₁#K₂)≥Cr(K₂) holds in general, here K₁#K₂ is the connected sum of K₁ and K₂ and Cr(K) stands for the crossing number of the link K. However, for alternating links K₁ and K₂, Cr(K₁#K₂)=Cr(K₁)+Cr(K₂) does hold. On the other hand, if K₁ is an alternating link and K₂ is any link, then we have Cr(K₁#K₂)≥Cr(K₁). In this paper, we show that there exists a wide class of links over which the crossing number is additive under the connected sum operation. This class is different from the class of all alternating links. It includes all torus knots and many alternating links. Furthermore, if K₁ is a connected sum of any given number of links from this class and K₂ is a non-trivial knot, we prove that Cr(K₁#K₂)≥Cr(K₁)+3.

The cubic lattice stick index of a knot type is the least number of sticks glued end-to-end that are necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p, p + 1)-torus knots, and show how composing and taking satellites can be used to obtain the cubic lattice stick index for a relatively large infinite class of knots. Additionally, we present several bounds relating cubic lattice stick index to other known invariants.

Twisted torus knots are torus knots with some full twists added along some number of adjacent strands. There are infinitely many known examples of twisted torus knots which are actually torus knots. We give eight more infinite families of such twisted torus knots with a single negative twist.

Using Kauffman's model of flat knotted ribbons, we demonstrate how all regular polygons of at least seven sides can be realized by ribbon constructions of torus knots. We calculate length to width ratios for these constructions thereby bounding the Ribbonlength of the knots. In particular, we give evidence that the closed (respectively, truncation) Ribbonlength of a (q + 1,q) torus knot is (2q + 1)cot(π/(2q + 1)) (respectively, 2qcot(π/(2q + 1))). Using these calculations, we provide the bounds c₁ ≤ 2/π and c₂ ≥ 5/3cotπ/5 for the constants c₁ and c₂ that relate Ribbonlength R(K) and crossing number C(K) in a conjecture of Kusner: c₁ C(K) ≤ R(K) ≤ c₂ C(K).

The Turaev genus and dealternating number of a link are two invariants that measure how far away a link is from alternating. We determine the Turaev genus of a torus knot with five or fewer strands either exactly or up to an error of at most one. We also determine the dealternating number of a torus knot with five or fewer strands up to an error of at most two. We also give bounds on the Turaev genus and dealternating number of torus links with five or fewer strands and on some infinite families of torus links on six strands.

A twisted torus knot is a torus knot with some consecutive strands twisted. More precisely, a twisted torus knot $T(p,q,r,s)$ is a torus knot $T(p,q)$ with $r$ consecutive strands $s$ times fully twisted. We determine which twisted torus knots $T(p,q,p-kq,-1)$ are a torus knot.

This paper is part expository and part presentation of calculational results. The target space of the Kontsevich integral for knots is a space of diagrams; this space has various algebraic structures which are described here. These are utilized with Le's theorem on the behaviour of the Kontsevich integral under cabling and with the Melvin-Morton Theorem, to obtain, in the Kontsevich integral for torus knots, both an explicit expression up to degree five and the general coefficients of the wheel diagrams.

A twisted torus knot $T(p,q,r,s)$ is a torus knot $T(p,q)$ with $r$ adjacent strands twisted fully $s$ times. In this paper, we determine the braid index of the knot $T(p,q,r,s)$ when the parameters $p,q,r$ satisfy $1<q<p<r\le p+q$. If the last parameter $s$ additionally satisfies $s=-1$, then we also determine the parameters $p,q,r$ for which $T(p,q,r,s)$ is a torus knot.

We address the concept of stick number for knots and links under various restrictions concerning the length of the sticks, the angles between sticks, and placements of the vertices. In particular, we focus on the effect of composition on the various stick numbers. Ultimately, we determine the traditional stick number for an infinite class of knots, which are the (n,n-1)-torus knots together with all of the possible compositions of such knots. The exact stick number was previously known for only seven knots.