Twist Knots Research Articles

Planar decomposition of the HOMFLY polynomial for bipartite knots and links

AbstractThe theory of the Kauffman bracket, which describes the Jones polynomial as a sum over closed circles formed by the planar resolution of vertices in a knot diagram, can be straightforwardly lifted from $$\mathfrak {sl}(2)$$ sl ( 2 ) to $$\mathfrak {sl}(N)$$ sl ( N ) at arbitrary N – but for a special class of bipartite diagrams made entirely from the antiparallel lock tangle. Many amusing and important knots and links can be described in this way, from twist and double braid knots to the celebrated Kanenobu knots for even parameters – and for all of them the entire HOMFLY polynomials possess planar decomposition. This provides an approach to evaluation of HOMFLY polynomials, which is complementary to the arborescent calculus, and this opens a new direction to homological techniques, parallel to Khovanov–Rozansky generalizations of the Kauffman calculus. Moreover, this planar calculus is also applicable to other symmetric representations beyond the fundamental one, and to links which are not fully bipartite what is illustrated by examples of Kanenobu-like links.

On the tau invariants in instanton and monopole Floer theories

AbstractWe unify two existing approaches to the tau invariants in instanton and monopole Floer theories, by identifying , defined by the second author via the minus flavors and of the knot homologies, with , defined by Baldwin and Sivek via cobordism maps of the 3‐manifold homologies induced by knot surgeries. We exhibit several consequences, including a relationship with Heegaard Floer theory, and use our result to compute and for twist knots.

Generalized torsion elements in the fundamental groups of 3-manifolds obtained by 0-surgeries along some double twist knots

We consider the 3-manifold obtained by the 0-surgery along a double twist knot. We construct a candidate for a generalized torsion element in the fundamental group of the surgered manifold, and see that there exists the cases where the candidate is actually a generalized torsion element. For a proof, we use the JSJ-decomposition of the surgered manifold. We also prove that the fundamental group of the 3-manifold obtained by 0-surgery along a double twist knot is bi-orderable if and only if it admits no generalized torsion elements. We also list some examples of the surgered manifolds whose fundamental groups admit generalized torsion elements.

Twisted Iwasawa invariants of knots

AbstractLet be a prime number and an integer coprime to . In the spirit of arithmetic topology, we introduce the notions of the twisted Iwasawa invariants of ‐representations and ‐covers of knots. We prove among other things that the set of Iwasawa invariants determines the genus and the fiberedness of a knot, yielding their profinite rigidity. Several intuitive examples are attached. We further prove the theorem for ‐representations of twist knot groups and give some remarks.

Geometric triangulations and the Teichmüller TQFT volume conjecture for twist knots

We construct a new infinite family of ideal triangulations and H–triangulations for the complements of twist knots, using a method originating from Thurston. These triangulations provide a new upper bound for the Matveev complexity of twist knot complements. We then prove that these ideal triangulations are geometric. The proof uses techniques of Futer and the second author, which consist in studying the volume functional on the polyhedron of angle structures. Finally, we use these triangulations to compute explicitly the partition function of the Teichmüller TQFT and to prove the associated volume conjecture for all twist knots, using the saddle point method.

Twisted magnetic knots and links

For magnetic knots and links in plasmas, we introduce an internal twist and study their dynamical behavior in numerical simulations. We use a set of helical and non-helical configurations and add an internal twist that cancels the helicity of the helical configurations or makes a non-helical system helical. These fields are then left to relax in a magnetohydrodynamic environment. In line with previous works, we confirm the importance of magnetic helicity in field relaxation. However, an internal twist, as could be observed in coronal magnetic loops, does not just add or subtract helicity, but also introduce an alignment of the magnetic field with the electric current, which is the source term for helicity. This source term is strong enough to lead to a significant change in magnetic helicity, which for some cases leads to a loss of the stabilizing properties expressed in the realizability condition. Even a relatively weak internal twist in these magnetic fields leads to a strong enough source term for magnetic helicity that for various cases even in a low diffusion environment, we observe an inversion in sign of magnetic helicity within timescales much shorter than the diffusion time. We conclude that in solar and stellar fields, an internal twist does not automatically result in a structurally stable configuration and that the alignment of the magnetic field with the electric current must be taken into account.

On the nonorientable 4-genus of double twist knots

We investigate the nonorientable 4-genus [Formula: see text] of a special family of 2-bridge knots, the double twist knots [Formula: see text]. Because the nonorientable 4-genus is bounded by the nonorientable 3-genus, it is known that [Formula: see text]. By using explicit constructions to obtain upper bounds on [Formula: see text] and known obstructions derived from Donaldson’s diagonalization theorem to obtain lower bounds on [Formula: see text], we produce infinite subfamilies of [Formula: see text] where [Formula: see text] and [Formula: see text], respectively. However, there remain infinitely many double twist knots where our work only shows that [Formula: see text] lies in one of the sets [Formula: see text], or [Formula: see text]. We tabulate our results for all [Formula: see text] with [Formula: see text] and [Formula: see text] up to 50. We also provide an infinite number of examples which answer a conjecture of Murakami and Yasuhara.

Explicit formulae for Chern-Simons invariants of the hyperbolic J(2n,-2m) knot orbifolds

We calculate the Chern-Simons invariants of the hyperbolic double twist knot orbifolds using the Schläfli formula for the generalized Chern-Simons function on the family of cone-manifold structures of double twist knots.

Unknotting twisted knots with Gauss diagram forbidden moves

Twisted knot theory, introduced by Bourgoin, is a generalization of virtual knot theory. It is well known that any virtual knot can be deformed into a trivial knot by a finite sequence of generalized Reidemeister moves and two forbidden moves [Formula: see text] and [Formula: see text]. Similarly, we show that any twisted knot also can be deformed into a trivial knot or a trivial knot with a bar by a finite sequence of extended Reidemeister moves and three forbidden moves [Formula: see text], [Formula: see text] (or [Formula: see text]) and [Formula: see text] (or [Formula: see text]).

Unexpected essential surfaces among exteriors of twisted torus knots

The twisted torus knots K(p, q; r, s) are obtained by performing a sequence of s full twists on r adjacent strands of (p, q)-torus knots. Morimoto asked whether all twisted torus knots with essential tori in the exterior fit into one of two families. We prove that the answer to this question is no, by finding two different new families of toroidal twisted torus knots.

Ribbonlength of families of folded ribbon knots

We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The folded ribbonlength is the length to width ratio of such a ribbon knot. We give upper bounds on the folded ribbonlength of 2-bridge, $(2,q)$ torus, twist, and pretzel knots, and these upper bounds turn out to be linear in the crossing number. We give a new way to fold $(p,q)$ torus knots and show that their folded ribbonlength is bounded above by $2p$. This means, for example, that the trefoil knot can be constructed with a folded ribbonlength of 6. We then show that any $(p,q)$ torus knot $K$ with $p\geq q>2$ has a constant $c>0$, such that the folded ribbonlength is bounded above by $c\cdot Cr(K)^{1/2}$. This provides an example of an upper bound on folded ribbonlength that is sub-linear in crossing number.

Evolution for Khovanov polynomials for figure-eight-like family of knots

We look at how evolution method deforms, when one considers Khovanov polynomials instead of Jones polynomials. We do this for the figure-eight-like knots (also known as ’double braid’ knots, see arXiv:1306.3197) — a two-parametric family of knots which “grows” from the figure-eight knot and contains both two-strand torus knots and twist knots. We prove that parameter space splits into four chambers, each with its own evolution, and two isolated points. Remarkably, the evolution in the Khovanov case features an extra eigenvalue, which drops out in the Jones [Formula: see text] limit.

Defect and degree of the Alexander polynomial

Defect characterizes the depth of factorization of terms in differential (cyclotomic) expansions of knot polynomials, i.e. of the non-perturbative Wilson averages in the Chern-Simons theory. We prove the conjecture that the defect can be alternatively described as the degree in q^{pm 2} of the fundamental Alexander polynomial, which formally corresponds to the case of no colors. We also pose a question if these Alexander polynomials can be arbitrary integer polynomials of a given degree. A first attempt to answer the latter question is a preliminary analysis of antiparallel descendants of the 2-strand torus knots, which provide a nice set of examples for all values of the defect. The answer turns out to be positive in the case of defect zero knots, what can be observed already in the case of twist knots. This proved conjecture also allows us to provide a complete set of C-polynomials for the symmetrically colored Alexander polynomials for defect zero. In this case, we achieve a complete separation of representation and knot variables.

A lower bound on the stable 4–genus of knots

We present a lower bound on the stable $4$-genus of a knot based on Casson-Gordon $\tau$-signatures. We compute the lower bound for an infinite family of knots, the twist knots, and show that a twist knot is torsion in the knot concordance group if and only if it has vanishing stable $4$-genus.

Totally geodesic surfaces in twist knot complements

In this article, we give explicit examples of infinitely many non-commensurable (non-arithmetic) hyperbolic $3$-manifolds admitting exactly $k$ totally geodesic surfaces for any positive integer $k$, answering a question of Bader, Fisher, Miller and Stover. The construction comes from a family of twist knot complements and their dihedral covers. The case $k=1$ arises from the uniqueness of an immersed totally geodesic thrice-punctured sphere, answering a question of Reid. Applying the proof techniques of the main result, we explicitly construct non-elementary maximal Fuchsian subgroups of infinite covolume within twist knot groups, and we also show that no twist knot complement with odd prime half twists is right-angled in the sense of Champanerkar, Kofman, and Purcell.

A stack-guiding unit constructed 2D COF with improved charge carrier transport and versatile photocatalytic functions

Two-dimensional COFs have shown great potential in photocatalysis, and the ordered layer arrangement is vital for the efficient utilization of photogenerated charge carriers. Herein, with a twisted knot of hexabenzocoronene, an ultrastable dioxin-linked 2D COF with undulated lattice was synthesized. As evidenced by N2 sorption, optoelectrochemical and transient absorption experiments and theoretical calculation, the concave-convex complementarity between layers effectively prevents stacking dislocation, prolongs the donor/acceptor π-columnar arrays and thus facilitates the transport of charge carriers. This COF was successfully applied to a series of photocatalytic reactions, such as the degradation of pollutants, the oxidative hydroxylation of arylboronic acids and the reduction of CO2. The activity exhibited by this COF in all the photocatalytic systems was obviously superior to the planar counterpart. The results of this work illustrate that the incorporation of stack-guiding building blocks is an effective strategy to improve the photocatalytic performance of COF materials.

A vanishing identity of adjoint Reidemeister torsions of twist knots

For a compact oriented 3-manifold with torus boundary the adjoint Reidemeister torsion is defined as a function on the $\mathrm{SL}_2(\mathbb{C})$-character variety depending on a choice of a boundary curve. Under reasonable assumptions, it is conjectured that the adjoint torsion satisfies a certain type of vanishing identities. In this paper, we prove that the conjecture holds for all hyperbolic twist knot exteriors by using Jacobi's residue theorem.

On two crossing numbers of algebraic knots under Hopf fibration

We answer a question posed by Fielder concerning two notions of crossing number for algebraic knots K under Hopf fibration. One notion, denoted h(K), is topological and the other, denoted Calg(K), comes from the realization of such knots around complex singularities. We show that Calg(K)−h(K) can be arbitrarily large. We also give an upper bound for h on some families of knots such as torus knots T(2,n), twist knots and their mirror images.

Torus knots obtained by negatively twisting torus knots

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.

Slope of orderable Dehn filling of two-bridge knots

In this paper, we study the Riley polynomial of double twist knots with higher genus. Using the root of the Riley polynomial, we compute the range of rational slope [Formula: see text] such that [Formula: see text]-filling of the knot complement has left-orderable fundamental group. Further more, we make a conjecture about left-orderable surgery slopes of two-bridge knots.