Torus Knots Research Articles

A REIDEMEISTER THEOREM FOR SOLID RIBBON TORUS LINKS

Abstract A complete Reidemeister characterisation of welded links is a long-standing open problem. We present a Reidemeister theorem for a related class of four-dimensional links: solid ribbon torus links, that is, immersed solid tori in $\mathbb {R}^4$ with only ribbon singularities, considered up to generalised ribbon isotopy.

Coloured Invariants of Torus Knots, "Equation missing" Algebras, and Relative Asymptotic Weight Multiplicities

Coloured Invariants of Torus Knots, "Equation missing" Algebras, and Relative Asymptotic Weight Multiplicities

The Jones polynomial for a torus knot with twists

We compute the Jones polynomial for a three-parameter family of links, the twisted torus links of the form T((p,q),(2,s)) where p and q are coprime and s is nonzero. When s=2n, these links are the twisted torus knots T(p,q;2,n). We show that for T(p,q;2,n), the Jones polynomial is trivial if and only if the knot is trivial.

An Elliptic Generalization of A1 Spherical DAHA at K = 2

Abstract We construct an algebra that is an elliptic generalization of $A_1$ spherical DAHA acting on its finite-dimensional module at $t=-q^{-K/2}$ with $K=2$. We prove that $PSL(2,\mathbb{Z})$ acts by automorphisms of the algebra we constructed, and provide an explicit representation of automorphisms and algebra operators alike by $3 \times 3$ matrices with matrix elements given by products of elliptic functions. A relation of this construction to the $K$-theory character of affine Laumon space is conjectured. We point out two potential applications, respectively to $SL(3,\mathbb{Z})$ symmetry of Felder–Varchenko functions and to new elliptic invariants of torus knots and Seifert manifolds.

A note on orientation-reversing distance one surgeries on non-null-homologous knots

We show that there are no distance one surgeries on non-null-homologous knots in M M that yield − M -M ( M M with opposite orientation) if M M is a 3-manifold obtained by a Dehn surgery on a knot K K in S 3 S^{3} , such that the order of its first homology is divisible by 9 9 but is not divisible by 27 27 . As an application, we show several knots, including the ( 2 , 9 ) (2,9) torus knot, do not have chirally cosmetic bandings. This simplifies the proof of a result first proven by Yang that the ( 2 , k ) (2,k) torus knot ( k > 1 ) (k>1) has a chirally cosmetic banding if and only if k = 5 k=5 .

Quandle coloring quivers of (p,q)-torus links

In 2019, Cho and Nelson introduced an enhancement of the quandle counting invariant, which is quandle coloring quivers. In 2021, Basi and Caprau used it to study [Formula: see text]-torus links. Recently, Zhou and Liu used it to study [Formula: see text]-torus links. In this paper, we study the quandle coloring space of [Formula: see text]- and [Formula: see text]-torus links by dihedral quandles and make a conjecture about the quandle coloring quiver of [Formula: see text]-torus links, where [Formula: see text] is an odd prime number.

Unknots, knots, links and necklaces made of dislocations in cholesterics

ABSTRACT We deal with the genesis and decay of knots and links made of dislocations in a cholesteric layer confined in the gap between orthogonal cylindrical mica sheets. The genesis starts by nucleation of folded unknots driven by a compressive strain. The nucleation process of the trivial and folded unknots is studied and discussed in detail. We show that the expansion of the folded unknots leads to their lateral collisions followed by the coalescence into the torus knots, such as the trefoil knot, or into the links, such as the Hopf and Solomon links. The viscoelastic decays of knots and links follow well-defined paths involving topological and geometrical transformations. In particular, the symmetric configuration of the Hopf link is unstable with respect to the shrinking of one of its interlaced loops. The final asymmetric configuration, made of a minimal loop tethered on a large cargo loop, is called the Hopf necklace. Similarly, the double Hopf necklace is obtained by the elastic decay of the Solomon link. We emphasise that Hopf necklaces of various orders are quite ubiquitous.

A note on surfaces in ℂℙ² and ℂℙ²#ℂℙ²

In this brief note, we investigate the C P 2 \mathbb {CP}^2 -genus of knots, i.e., the least genus of a smooth, compact, orientable surface in C P 2 ∖ B 4 ˚ \mathbb {CP}^2\smallsetminus \mathring {B^4} bounded by a knot in S 3 S^3 . We show that this quantity is unbounded, unlike its topological counterpart. We also investigate the C P 2 \mathbb {CP}^2 -genus of torus knots. We apply these results to improve the minimal genus bound for some homology classes in C P 2 # C P 2 \mathbb {CP}^2\# \mathbb {CP} ^2 .

Efficiency of non-identical double helix patterns in minimizing ropelength of torus knots

The ropelength of a knotted string with volume is defined as the ratio of the length of its central curve to the radius of its sectional disc. In a physical context, achieving minimal ropelength corresponds to a state of minimal potential energy, and geometrically, it signifies a tightly-packed conformation. The quest to establish a connection between the topological complexity of knotted strings and their minimal ropelength has persisted into recent years. In this paper, we introduce a new upper bound on the minimal ropelength of (2, n)-torus knots and links: Rop(T(2, n)) ≤ 7.3163 Cr(T(2, n)) + 17.1657. This upper bound is derived from a torus knot conformation constructed based on a tightened pattern of double helix with non-identical radii of winding. A comparative analysis with conformations generated from a superhelix and a circular helix underscores the efficiency of the non-identical double helix pattern, particularly when it appears as a long repeated motif in knotted strings.

Characters of logarithmic vertex operator algebras and coloured invariants of torus links

We show that the characters of s l r \mathfrak {sl}_r versions of the ( 1 , p ) (1,p) singlet and the ( 1 , p ) (1,p) triplet vertex operator algebras arise as limits of appropriately coloured s l r \mathfrak {sl}_r Jones invariants of certain torus links.

Brieskorn spheres, cyclic group actions and the Milnor conjecture

AbstractIn this paper we further develop the theory of equivariant Seiberg–Witten–Floer cohomology of the two authors, with an emphasis on Brieskorn homology spheres. We obtain a number of applications. First, we show that the knot concordance invariants defined by the first author satisfy for torus knots, whenever is a prime not dividing . Since is a lower bound for the slice genus, this gives a new proof of the Milnor conjecture. Second, we prove that a free cyclic group action on a Brieskorn homology 3‐sphere does not extend smoothly to any homology 4‐ball bounding . In the case of a non‐free cyclic group action of prime order, we prove that if the rank of is greater than times the rank of , then the ‐action on does not extend smoothly to any homology 4‐ball bounding . Third, we prove that for all but finitely many primes a similar non‐extension result holds in the case that the bounding 4‐manifold has positive‐definite intersection form. Finally, we also prove non‐extension results for equivariant connected sums of Brieskorn homology spheres.

Torus knot designs using a Rubik’s snake

A Rubik’s Snake is a toy that was invented over 40 years ago together with the more famous Rubik’s Cube. It can be twisted to many interesting shapes including knots. In this paper, we study Rubik’s Snake torus knot designs. The general solutions are given for all torus knots T(2,n), T(3,n) and T(4,n). Some more challenging constructions are also provided.

Torus knot filtered embedded contact homology of the tight contact 3‐sphere

AbstractKnot filtered embedded contact homology was first introduced by Hutchings in 2015; it has been computed for the standard transverse unknot in irrational ellipsoids by Hutchings and for the Hopf link in lens spaces via a quotient by Weiler. While toric constructions can be used to understand the ECH chain complexes of many contact forms adapted to open books with binding the unknot and Hopf link, they do not readily adapt to general torus knots and links. In this paper, we generalize the definition and invariance of knot filtered embedded contact homology to allow for degenerate knots with rational rotation numbers. We then develop new methods for understanding the embedded contact homology chain complex of positive torus knotted fibrations of the standard tight contact 3‐sphere in terms of their presentation as open books and as Seifert fiber spaces. We provide Morse–Bott methods, using a doubly filtered complex and the energy filtered perturbed Seiberg–Witten Floer theory developed by Hutchings and Taubes, and use them to compute the knot filtered embedded contact homology, for odd and positive.

Harer–Zagier transform of the HOMFLY–PT polynomial for families of twisted hyperbolic knots

In an attempt to generalise knot matrix models for non-torus knots, which currently remains an open problem, we derived expressions for the Harer–Zagier transform—a discrete Laplace transform—of the HOMFLY–PT polynomial for some infinite families of twisted hyperbolic knots. Among them, we found a family of pretzel knots for which, like for torus knots, the transform has a fully factorised form, while for the remaining families considered it consists of sums of factorised terms. Their zero loci show a remarkable structure, which mostly lies on the unit circle and deviates from it only in pairs.

Twisted Blanchfield pairings and twisted signatures III: Applications

Abstract This paper describes how to compute algorithmically certain twisted signature invariants of a knot $K$ using twisted Blanchfield forms. An illustration of the algorithm is implemented on $(2,q)$ -torus knots. Additionally, using satellite formulas for these invariants, we also show how to obstruct the sliceness of certain iterated torus knots.

Supervariable approach to particle on a torus knot: a model for Hodge theory

Supervariable approach to particle on a torus knot: a model for Hodge theory

Instantons and Khovanov skein homology on $I\times T^{2}$

Asaeda, Przytycki, and Sikora (2004) defined a generalization of Khovanov homology for links in I -bundles over compact surfaces. We prove that, for a link L\subset (-1,1)\times T^{2} , the Asaeda–Przytycki–Sikora homology of L has rank 2 with \mathbb{Z}/2 -coefficients if and only if L is isotopic to an embedded knot in \{0\}\times T^{2} . We also prove that Asaeda–Przytycki–Sikora homology detects the unlink and torus links in (-1,1)\times T^{2} .

Geometric realisation of relation modules over aspherical groups

We prove that the direct sums of extensions of scalars of relation modules are geometrically realisable as the second homotopy group of a finite 2-complex. We use this to exhibit a finite 2-complex with fundamental group the (10,15) torus knot group and non-free π2, yielding exotic presentations of a group for which no such examples had previously been known. We conclude by constructing stably free non-free modules over an infinite family of Baumslag-Solitar groups; it remains to determine whether these modules are geometrically realisable by finite 2-complexes.

Trisections induced by the Gluck surgery along certain spun knots

Gay and Meier asked whether or not a trisection diagram constructed by the Gluck twist on a spun or a twist spun 2-knot obtained from some method is standard. In this paper, we depict the trisection diagrams explicitly when the 2-knot is the spun [Formula: see text]-torus knot, where [Formula: see text], and show that the trisection diagram is standard when [Formula: see text]. Moreover, we introduce a notion of homologically standard for trisection diagrams and show that the trisection diagram is homologically standard for all [Formula: see text].

Topological entanglement entropy for torus-knot bipartitions and the Verlinde-like formulas

The topological Rényi and entanglement entropies depend on the bipartition of the manifold and the choice of the ground states. However, these entanglement quantities remain invariant under a coordinate transformation when the bipartition also undergoes the identical transformation. In topological quantum field theories (TQFTs), these coordinate transformations reduce to representations of the mapping class group on the manifold of the Hilbert space. We employ this invariant property of the Rényi and entanglement entropies under coordinate transformations for TQFTs in (2 + 1) dimensions on a torus with various bipartitions. By utilizing the replica trick and the surgery method to compute the topological Rényi and entanglement entropies, the invariant property results in Verlinde-like formulas. Furthermore, for the bipartition with interfaces as two non-intersecting torus knots, an SL(2, ℤ) transformation can untwist the torus knots, leading to a simple bipartition with an effective ground state. This invariant property allows us to demonstrate that the topological entanglement entropy has a lower bound −2 ln D, where D is the total quantum dimensions of the system.