Torus Knots Research Articles

Gluing formulas for the L2-Alexander torsions

We prove a Torres-like formula for the [Formula: see text]-Alexander torsions of links, as well as formulas for connected sums and cablings of links. Along the way we compute explicitly the [Formula: see text]-Alexander torsions of torus links inside the three-sphere, the solid torus and the thickened torus.

Conformal Villarceau Rotors

We consider Villarceau circles as the orbits of specific composite rotors in 3D conformal geometric algebra that generate knots on nested tori. We compute the conformal parametrization of these circular orbits by giving an equivalent, position-dependent simple rotor that generates the same parametric track for a given point. This allows compact derivation of the quantitative symmetry properties of the Villarceau circles. We briefly derive their role in the Hopf fibration and as stereographic images of isoclinic rotations on a 3-sphere of the 4D Clifford torus. We use the CGA description to generate 3D images of our results, by means of GAviewer. This paper was motivated by the hope that the compact coordinate-free CGA representations can aid in the analysis of Villarceau circles (and torus knots) as occurring in the Maxwell and Dirac equations.

Nontrivial Elements in a Knot Group That are Trivialized by Dehn Fillings

AbstractLet $K$ be a nontrivial knot in $S^3$ with the exterior $E(K)$, and $\gamma \in G(K) = \pi _1(E(K), *)$ a slope element represented by an essential simple closed curve on $\partial E(K)$ with base point $* \in \partial E(K)$. Since the normal closure $\langle \!\langle \gamma \rangle \!\rangle $ of $\gamma $ in $G(K)$ coincides with that of $\gamma ^{-1}$, and $\gamma $ and $\gamma ^{-1}$ correspond to a slope $r \in \mathbb{Q} \cup \{ \infty \}$, we write $\langle \!\langle r \rangle \!\rangle = \langle \!\langle \gamma \rangle \!\rangle $. The normal closure $\langle \!\langle r \rangle \!\rangle $ describes elements, which are trivialized by $r$-Dehn filling of $E(K)$. In this article, we prove that $\langle \!\langle r_1 \rangle \!\rangle = \langle \!\langle r_2 \rangle \!\rangle $ if and only if $r_1 = r_2$, and for a given finite family of slopes $\mathcal{S} = \{ r_1, \dots , r_n \}$, the intersection $\langle \!\langle r_1 \rangle \!\rangle \cap \cdots \cap \langle \!\langle r_n \rangle \!\rangle $ contains infinitely many elements except when $K$ is a $(p, q)$-torus knot and $pq \in \mathcal{S}$. We also investigate inclusion relation among normal closures of slope elements.

Incoherent nullification of torus knots and links

In the paper, we study the incoherent nullification number [Formula: see text] of knots and links. We establish an upper bound on the incoherent nullification number of torus knots and links and conjecture that this upper bound is the actual incoherent nullification number of this family. Finally, we establish the actual incoherent nullification number of particular subfamilies of torus knots and links.

Brieskorn Manifolds, Generalized Sieradski Groups, and Coverings of Lens Spaces

The Brieskorn manifolds $B(p,q,r)$ are the $r$-fold cyclic coverings of the 3-sphere $S^{3}$ branched over the torus knot $T(p,q)$. The generalised Sieradski groups $S(m,p,q)$ are groups with $m$-cyclic pre\-sen\-tation $G_{m}(w)$, where defining word $w$ has a special form, depending of $p$ and $q$. In particular, $S(m,3,2) = G_{m}(w)$ is the group with $m$ generators $x_{1}, \ldots, x_{m}$ and $m$ defining relations $w(x_{i}, x_{i+1}, x_{i+2})=1$, where $w(x_{i}, x_{i+1}, x_{i+2}) = x_{i} x_{i+2} x_{i+1}^{-1}$. Presentations of $S(2n,3,2)$ in a certain form $G_{n}(w)$ were investigated by Howie and Williams. They proved that the $n$-cyclic presentations are geometric, i.e., correspond to the spines of closed orientable 3-manifolds. We establish an analogous result for the groups $S(2n,5,2)$. It is shown that in both cases the manifolds are $n$-fold cyclic branched coverings of lens spaces. To classify some of constructed manifolds we used the Matveev's computer program "Recognizer".

Distinguishing the generalized knot groups of square and granny knot analogues

Given a knot [Formula: see text], we may construct a group [Formula: see text] from the fundamental group of [Formula: see text] by adjoining an [Formula: see text]th root of the meridian that commutes with the corresponding longitude. For [Formula: see text] these “generalized knot groups” determine [Formula: see text] up to reflection. The second author has shown that for [Formula: see text], 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 [Formula: see text], [Formula: see text], constructed as connected sums of [Formula: see text]-torus knots of opposite or identical chiralities. More precisely, for coprime [Formula: see text] and [Formula: see text] satisfying a coprimality condition with [Formula: see text] and [Formula: see text], we construct an explicit finite group [Formula: see text] (depending on [Formula: see text], [Formula: see text] and [Formula: see text]) such that [Formula: see text] and [Formula: see text] can be distinguished by counting homomorphisms into [Formula: see text]. The coprimality condition includes all [Formula: see text] coprime to [Formula: see text]. The result shows that the difference between these two groups can be detected using a finite group.

Finite n-quandles of torus and two-bridge links

We compute Cayley graphs and automorphism groups for all finite [Formula: see text]-quandles of two-bridge and torus knots and links, as well as torus links with an axis.

Positively twisted torus knots which are torus knots

A twisted torus knot [Formula: see text] is obtained from a torus knot [Formula: see text] by twisting [Formula: see text] adjacent strands of [Formula: see text] fully [Formula: see text] times. In this paper, we determine the parameters [Formula: see text] for which [Formula: see text] is a torus knot with [Formula: see text].

The twisted Reidemeister torsion of an iterated torus knot

We calculate the twisted Reidemeister torsion of the complement of an iterated torus knot associated with a representation of its fundamental group to the complex special linear group of degree two. We also show that the twisted Reidemeister torsions associated with various representations appear in the asymptotic expansion of the colored Jones polynomial.

Dynamics of linked filaments in excitable media

In this paper we present the results of parallel numerical computations of the long-term dynamics of linked vortex filaments in a three-dimensional FitzHugh–Nagumo excitable medium. In particular, we study all torus links with no more than 12 crossings and identify a timescale over which the dynamics is regular in the sense that each link is well-described by a spinning rigid conformation of fixed size that propagates at constant speed along the axis of rotation. We compute the properties of these links and demonstrate that they have a simple dependence on the crossing number of the link for a fixed number of link components. Furthermore, we find that instabilities that exist over longer timescales in the bulk can be removed by boundary interactions that yield stable torus links which settle snugly at the medium boundary. The Borromean rings are used as an example of a non-torus link to demonstrate both the irregular tumbling dynamics that arises in the bulk and its suppression by a tight confining medium. Finally, we investigate the collision of torus links and reveal that this produces a complicated wrestling motion where one torus link can eventually dominate over the other by pushing it into the boundary of the medium.

Quantum link homology via trace functor I

Motivated by topology, we develop a general theory of traces and shadows for an endobicategory, which is a pair: bicategory and endobifunctor . For a graded linear bicategory and a fixed invertible parameter q, we quantize this theory by using the endofunctor $$\Sigma _q$$ such that $$\Sigma _q \alpha :=q^{-\deg \alpha }\Sigma \alpha $$ for any 2-morphism $$\alpha $$ and coincides with $$\Sigma $$ otherwise. Applying the quantized trace to the bicategory of Chen–Khovanov bimodules we get a new triply graded link homology theory called quantum annular link homology. If $$q=1$$ we reproduce Asaeda–Przytycki–Sikora homology for links in a thickened annulus. We prove that our homology carries an action of , which intertwines the action of cobordisms. In particular, the quantum annular homology of an n-cable admits an action of the braid group, which commutes with the quantum group action and factors through the Jones skein relation. This produces a nontrivial invariant for surfaces knotted in four dimensions. Moreover, a direct computation for torus links shows that the rank of quantum annular homology groups depend on the quantum parameter q.

Left-orderability for surgeries on twisted torus knots

We show that the fundamental group of the 3-manifold obtained by $\frac{p}{q}$-surgery along the $(n-2)$-twisted $(3,3m+2)$-torus knot, with $n,m \ge 1$, is not left-orderable if $\frac{p}{q} \ge 2n + 6m-3$ and is left-orderable if $\frac{p}{q}$ is sufficiently close to 0.

On sums of torus knots concordant to alternating knots

We consider the question, asked by Friedl, Livingston and Zentner, of which sums of torus knots are concordant to alternating knots. After a brief analysis of the problem in its full generality, we focus on sums of two torus knots. We describe some effective obstructions based on Heegaard Floer homology.

Representations, sheaves and Legendrian (2,m) torus links

We study an $A_\infty$ category associated to Legendrian links in $\mathbb{R}^3$ whose objects are $n$-dimensional representations of the Chekanov-Eliashberg differential graded algebra of the link. This representation category generalizes the positive augmentation category and we conjecture that it is equivalent to a category of sheaves of microlocal rank $n$ constructed by Shende, Treumann, and Zaslow. We establish the cohomological version of this conjecture for a family of Legendrian $(2,m)$ torus links.

A note on Dehn colorings and invariant factors

If [Formula: see text] is an abelian group and [Formula: see text] is an integer, let [Formula: see text] be the subgroup of [Formula: see text] consisting of elements [Formula: see text] such that [Formula: see text]. We prove that if [Formula: see text] is a diagram of a classical link [Formula: see text] and [Formula: see text] are the invariant factors of an adjusted Goeritz matrix of [Formula: see text], then the group [Formula: see text] of Dehn colorings of [Formula: see text] with values in [Formula: see text] is isomorphic to the direct product of [Formula: see text] and [Formula: see text]. It follows that the Dehn coloring groups of [Formula: see text] are isomorphic to those of a connected sum of torus links [Formula: see text].

On the computation of torus link homology

We introduce a new method for computing triply graded link homology, which is particularly well adapted to torus links. Our main application is to the$(n,n)$-torus links, for which we give an exact answer for all$n$. In several cases, our computations verify conjectures of Gorskyet al.relating homology of torus links with Hilbert schemes.

Modules over plane curve singularities in any ranks and DAHA

We generalize the construction of geometric superpolynomials for unibranch plane curve singularities from our prior paper from rank one to any ranks; explicit formulas are obtained for torus knots. The new feature is the definition of counterparts of Jacobian factors (directly related to compactified Jacobians) for higher ranks, which is parallel to the classical passage from invertible sheaves to vector bundles over algebraic curves. This is an entirely local theory, connected with affine Springer fibers for non-reduced (germs of) spectral curves. We conjecture and justify numerically the connection of our geometric polynomials in arbitrary ranks with the corresponding DAHA superpolynomials for any algebraic knots colored by columns.

On the complexity of torus knot recognition

We show that the problem of recognizing that a knot diagram represents a specific torus knot, or any torus knot at all, is in the complexity class ${\sf NP} \cap {\sf co\text{-}NP}$, assuming the generalized Riemann hypothesis. We also show that satellite knot detection is in ${\sf NP}$ under the same assumption, and that cabled knot detection and composite knot detection are unconditionally in ${\sf NP}$. Our algorithms are based on recent work of Kuperberg and of Lackenby on detecting knottedness.

Independence of iterated Whitehead doubles

A theorem of Furuta and Fintushel-Stern provides a criterion for a collection of Seifert fibred homology spheres to be independent in the homology cobordism group of oriented homology 3-spheres. In this article we use these results and some 4-dimensional constructions to produce infinite families of positive torus knots whose iterated Whitehead doubles are independent in the smooth concordance group.

Ropelength of superhelices and (2, n)-torus knots

In this paper we investigate a ropelength-minimizing conformation of 4-strand superhelical strings whose axial curves constitute the standard double helix.In Huh et al (2016 J. Phys. A: Math. Theor. 49 415205) the authors found a specific conformation of standard double helix which was mathematically shown to be the unique ropelength-minimizing conformation. Adopting the conformation as axial curves we present a parametrization of superhelical curves so that the resulting shape is controlled by the twist number N and the helical radius r2. For each N, the value of r2 minimizing the ropelength is numerically estimated. A notable observation is that the ropelength per crossing of our 4-strand superhelical strings is minimized around 2.96 < N < 2.97 which suggests a moment of transition from tightly-packed status to unpacked status.As an application of our estimation, we derive an upper bound on the ropelength of -torus knots, which is 45.8237k + 28.4223. Finally the efficiency of our superhelix model for -torus knots is discussed in comparison with the circular helix model.