Narrow Results

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.

Rudimentary knots are invoked to generate a representation of the elementary particles, a model that endows the particles with visualizable structure. The model correlates with the basic tenets, taxonomy, and interactions of the Standard Model, but goes beyond it in a number of important ways, the most significant being that all particles (hadrons and leptons, fermions and bosons) and interactions share a common topology. Among other consequences of the modeling are the topological basis for isospin invariance and its connection to electric charge, the necessary identity of electron and proton charge magnitudes, and the existence of precisely three generations on the particle family tree. The salient feature of the model is that the elementary particles are viewed not as discrete, point-like objects in a vacuum but rather as sustainable, membrane-like distortions embodying curvature and torsion in and of an otherwise featureless continuum and that their manifest physical attributes correlate with the distortion. There are additional connections to the theories of fiber bundles, superstrings and instantons and, historically, to the work of Kelvin in the mid-nineteenth century and Cartan in the 1920s among others.

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).

We show that no torus knot of type (2, n), n > 3 odd, can be obtained from a polynomial embedding t ↦ (f(t), g(t), h(t)) where (deg(f), deg(g)) ≤ (3, n + 1). Eventually, we give explicit examples with minimal lexicographic degree.

The stick index of a knot K is defined to be the least number of line segments needed to construct a polygonal embedding of K. We define the projection stick index of K to be the least number of line segments in any projection of a polygonal embedding of K. In this paper, we establish bounds on the projection stick index for various torus knots. We then show that the stick index of a (p, 2p + 1)-torus knot is 4p, and the projection stick index is 2p + 1. This provides examples of knots such that the projection stick index is one greater than half the stick index. We show that for all other torus knots for which the stick index is known, the projection stick index is larger than this. We conjecture that a projection stick index of half the stick index is unattainable for any knot.

Colorings of torus knots and twist-spun torus knots by general Alexander quandles over finite fields are explicitly determined. It is then observed that, for some standard Mochizuki's cocycles, a large class of twist-spun torus knots with non-trivial colorings have trivial CJKLS state-sum invariants.

In this paper we resolve some conjectures concerning positive braid knots and almost alternating torus knots. Namely, we prove that the first Khovanov homology group of positive braid knot is trivial, as conjectured by Khovanov. Also, we generalize this result to show that the same is true in the case of Khovanov–Rozansky homology (sl(n) link homology) for any positive integer n. Moreover, by using the Khovanov homology theory, we prove the classical knot theory conjecture by Adams, that the only almost alternating torus knots are T_{3, 4} and T_{3, 5}.

The subject is a localized disturbance in the form of a torus knot of an otherwise featureless continuum. The knot's topologically quantized, self-sustaining nature emerges in an elementary, straightforward way on the basis of a simple geometric model, one that constrains the differential geometric basis it otherwise shares with General Relativity (GR). Two approaches are employed to generate the knot's solitonic nature, one emphasizing basic differential geometry and the other based on a Lagrangian. The relationship to GR is also examined, especially in terms of the formulation of an energy density for the Lagrangian. The emergent knot formalism is used to derive estimates of some measurable quantities for a certain elementary particle model documented in previous publications. Also emerging is the compatibility of the torus knot formalism and, by extension, that of the cited particle model, with general relativity as well as with the Dirac theoretic notion of antiparticles.

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.

For each prime p > 7 we obtain the expression for an upper bound on the minimum number of colors needed to non-trivially color T(2, p), the torus knot of type (2, p), modulo p. This expression is t + 2l -1 where t and l are extracted from the prime p. It is obtained from iterating the so-called Teneva transformations which we introduced in a previous article. With the aid of our estimate we show that the ratio "number of colors needed vs. number of colors available" tends to decrease with increasing modulus p. For instance as of prime 331, the number of colors needed is already one tenth of the number of colors available. Furthermore, we prove that 5 is the minimum number of colors needed to non-trivially color T(2, 11) modulo 11. Finally, as a preview of our future work, we prove that 5 is the minimum number of colors modulo 11 for two rational knots with determinant 11.

Region crossing change for a knot or a proper link is an unknotting operation. In this paper, we provide a sharp upper bound on the region unknotting number for a large class of torus knots and proper links. Also, we discuss conditions on torus links to be proper.

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.

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 $T(p,q,r,s)$ 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 $s>0$.

We compute Cayley graphs and automorphism groups for all finite $n$-quandles of two-bridge and torus knots and links, as well as torus links with an axis.

Given a knot $K$, we may construct a group $G_n(K)$ from the fundamental group of $K$ by adjoining an $n$th root of the meridian that commutes with the corresponding longitude. For $n\ge 2$ these “generalized knot groups” determine $K$ up to reflection.

The second author has shown that for $n\ge 2$, the generalized knot groups of the square and granny knots can be distinguished by counting homomorphisms into a suitably chosen finite group. We extend this result to certain generalized knot groups of square and granny knot analogues $S{K}_{a,b}=T_{a,b}\#T_{-a,b}$, $G{K}_{a,b}=T_{a,b}\#T_{a,b}$, constructed as connected sums of $(a,b)$-torus knots of opposite or identical chiralities. More precisely, for coprime $a,b\ge 2$ and $n$ satisfying a coprimality condition with $a$ and $b$, we construct an explicit finite group $G$ (depending on $a$, $b$ and $n$) such that $G_n(S{K}_{a,b})$ and $G_n(G{K}_{a,b})$ can be distinguished by counting homomorphisms into $G$. The coprimality condition includes all $n\ge 2$ coprime to $ab$. The result shows that the difference between these two groups can be detected using a finite group.

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.

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.

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.

Consider the knots obtained from torus knots by adding a negative full twist along two adjacent strands. Among these knots, we determine which are cable knots or torus knots.