Figure-eight Knot Research Articles

The $(2,1)$-cable of the figure-eight knot is not smoothly slice

The $(2,1)$-cable of the figure-eight knot is not smoothly slice

A conjecture of Zagier and the value distribution of quantum modular forms

In his influential paper on quantum modular forms, Zagier developed a conjectural framework describing the behavior of certain quantum knot invariants under the action of the modular group on their arguments. More precisely, when J_{K,0} denotes the colored Jones polynomial of a knot K , Zagier’s modularity conjecture describes the asymptotics of the quotient J_{K,0}(e^{2 \pi i \gamma(x)}) / J_{K,0}(e^{2 \pi i x}) as x \to \infty along rationals with bounded denominators, where \gamma \in \mathrm{SL}(2,\mathbb{Z}) . This problem is most accessible for the figure-eight knot 4_{1} , where the colored Jones polynomial has a simple explicit expression in terms of the q -Pochhammer symbol. Zagier also conjectured that the function h(x) = \log (J_{4_1,0}(e^{2 \pi i x}) / J_{4_1,0}(e^{2 \pi i /x})) can be extended to a function on \mathbb{R} which is continuous at irrationals. In the present paper, we prove Zagier’s continuity conjecture for all irrationals for which the sequence of partial quotients in the continued fraction expansion is unbounded. In particular, the continuity conjecture holds almost everywhere on the real line. We also establish a smooth approximation of h , uniform over all rationals, in accordance with the modularity conjecture. As an application, we find the limit distribution (after a suitable centering and rescaling) of \log J_{4_1,0}(e^{2 \pi i x}) , when x ranges over all reduced rationals in (0,1) with denominator at most N , as N \to \infty , thereby confirming a conjecture of Bettin and Drappeau.

Switchable mode-locked ytterbium-doped fiber laser based on macrobending loss tuning of figure-eight knot fiber structure

Switchable mode-locked ytterbium-doped fiber laser based on macrobending loss tuning of figure-eight knot fiber structure

Adjoint Reidemeister torsions of some 3-manifolds obtained by Dehn surgeries

Abstract We determine the adjoint Reidemeister torsion of a $3$ -manifold obtained by some Dehn surgery along K, where K is either the figure-eight knot or the $5_2$ -knot. As in a vanishing conjecture (Benini et al. (2020, Journal of High Energy Physics 2020, 57), Gang et al. (2020, Journal of High Energy Physics 2020, 164), and Gang et al. (2021, Advances in Theoretical and Mathematical Physics 25, 1819–1845)), we consider a similar conjecture and show that the conjecture holds for the 3-manifold.

Capsizing due to friction-induced twist in the failure of stopper knots

We investigate the failure mechanism of stopper knots, with a particular focus on the figure-8 knot as a representative example. Stopper knots are widely used in climbing, sailing, racket stringing, and sewing to maintain tension in ropes, strings, or threads while preventing them from passing through an orifice. Combining high-precision model experiments and Finite Element Analyses, we systematically explore the influence of frictional interactions and their role in the build-up of mechanical twist. Our findings reveal that the failure of stopper knots via capsizing, which involves configurational alterations of the filament, is primarily due to friction-induced twisting when loading the knot against a restraining plate containing a clearance hole. Our study offers a comprehensive understanding of the mechanical behavior of stopper knots under diverse loading conditions, thereby providing crucial insights for their reliable application across various domains.

Bounds in simple hexagonal lattice and classification of 11-stick knots

The stick number and the edge length of a knot type in the simple hexagonal lattice (sh-lattice) are the minimal numbers of sticks and edges required, respectively, to construct a knot of the given type in sh-lattice. By introducing a linear transformation between lattices, we prove that for any given knot both values in the sh-lattice are strictly less than the values in the cubic lattice. Finally, we show that the only non-trivial [Formula: see text]-stick knots in the sh-lattice are the trefoil knot ([Formula: see text]) and the figure-eight knot ([Formula: see text]).

Anosov flows on Dehn surgeries on the figure-eight knot

Anosov flows on Dehn surgeries on the figure-eight knot

Change in grip strength, hang time, and knot tying speed after 24 hours of endurance rock climbing.

Non-professional climbers are increasingly attempting long routes in a single day. Many suffer injury or rely on search and rescue teams when they become too fatigued to finish. Predicting fatigue is difficult, and existing studies have only studied climbers over durations less than an hour, while many outdoor multipitch climbs require more than an hour of climbing. To determine how strength, endurance, and dexterity reflect fatigue after 24 h of continuous climbing. Volunteer competitors completed measurements of grip strength, static hang time to failure, and time to tie a figure-eight follow-through knot. Measurements were taken during the registration period before the competition and again within an hour after the competition ended. Measurements were compared using the paired t-test. Subgroup analysis was applied to competitors by division. Linear regression was applied to determine the relationship between vertical feet climbed and the number of routes climbed during the competition on each metric. Thirty-six total climbers (average age 29.4 years old) completed pre- and post-competition measurements. After 24 h of climbing (n = 36), mean grip strength decreased by 14.3-15 lbs or 14.7%-15.1% (p < 0.001) and static hang time decreased by 54.2 s or 71.2% (p < 0.001). There was no significant change in time to tie a figure-eight-follow-through knot. Grip strength and hang time decreases were significant in climbers with outdoor redpoints of 5.10a and above. Hang time decreased by 5.4 s per 1,000 vertical feet climbed (p = 0.044). Climbers can expect to experience a 14.7%-15.1% decrease in grip strength and 71.2% decrease in static hang time after 24 h of continuous climbing. These changes may make it difficult to climb consistently over a long objective, and climbers can use these measures at home to train for longer climbing routes. Future studies on shorter climbing intervals can help determine rates of decline in performance measures.

The colored Jones polynomial of a cable of the figure-eight knot

We study the asymptotic behavior of the [Formula: see text]-dimensional colored Jones polynomial of a cable of the figure-eight knot, evaluated at [Formula: see text] for a real number [Formula: see text]. We show that if [Formula: see text] is sufficiently large, the colored Jones polynomial grows exponentially when [Formula: see text] goes to the infinity. Moreover the growth rate is related to the Chern–Simons invariant of the knot exterior associated with an [Formula: see text] representation.

Limit Laws of Maximal Birkhoff Sums for Circle Rotations via Quantum Modular Forms

Abstract In this paper, we show how quantum modular forms naturally arise in the ergodic theory of circle rotations. Working with the classical Birkhoff sum $S_{N}(\alpha )=\sum _{n=1}^{N} (\{ n \alpha \}-1/2)$, we prove that the maximum and the minimum as well as certain exponential moments of $S_{N}(r)$ as functions of $r \in \mathbb{Q}$ satisfy a direct analogue of Zagier’s continuity conjecture, originally stated for a quantum invariant of the figure-eight knot. As a corollary, we find the limit distribution of $\max _{0 \le N&amp;lt;M} S_{N}(\alpha )$ and $\min _{0 \le N&amp;lt;M} S_{N}(\alpha )$ with a random $\alpha \in [0,1]$.

Maximizing Sudler products via Ostrowski expansions and cotangent sums

There is an extensive literature on the asymptotic order of Sudler's trigonometric product $P_N (\alpha) = \prod_{n=1}^N |2 \sin (\pi n \alpha)|$ for fixed or for "typical" values of $\alpha$. In the present paper we establish a structural result, which for a given $\alpha$ characterizes those $N$ for which $P_N(\alpha)$ attains particularly large values. This characterization relies on the coefficients of $N$ in its Ostrowski expansion with respect to $\alpha$, and allows us to obtain very precise estimates for $\max_{1 \le N \leq M} P_N(\alpha)$ and for $\sum_{N=1}^M P_N(\alpha)^c$ in terms of $M$, for any $c>0$. Furthermore, our arguments give a natural explanation of the fact that the value of the hyperbolic volume of the complement of the figure-eight knot appears generically in results on the asymptotic order of the Sudler product and of the Kashaev invariant.

Strict inequalities for the n-crossing number

In 2013, Adams introduced the [Formula: see text]-crossing number of a knot [Formula: see text], denoted by [Formula: see text]. Inequalities between the [Formula: see text]-, [Formula: see text]-, [Formula: see text]- and [Formula: see text]-crossing numbers have been previously established. We prove [Formula: see text] for all knots [Formula: see text] that are not the trivial, trefoil, or figure-eight knot. We show this inequality is optimal and obtain previously unknown values for [Formula: see text].

The colored Jones polynomial of the figure-eight knot and a quantum modularity

AbstractWe study the asymptotic behavior of the N-dimensional colored Jones polynomial of the figure-eight knot evaluated at $\exp \bigl ((u+2p\pi \sqrt {-1})/N\bigr )$ , where u is a small real number and p is a positive integer. We show that it is asymptotically equivalent to the product of the p-dimensional colored Jones polynomial evaluated at $\exp \bigl (4N\pi ^2/(u+2p\pi \sqrt {-1})\bigr )$ and a term that grows exponentially with growth rate determined by the Chern–Simons invariant. This indicates a quantum modularity of the colored Jones polynomial.

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.

Hidden symmetries of hyperbolic links

W. D. Neumann and A. W. Reid conjectured that the figure-eight knot and the two dodecahedral knots are the only hyperbolic knots admitting hidden symmetries. We construct an $n$-component hyperbolic link whose complement admits hidden symmetries for any $n\ge 4$.

Profinite Rigidity, Kleinian Groups, and the Cofinite Hopf Property

Let Γ be a nonelementary Kleinian group and H<Γ be a finitely generated, proper subgroup. We prove that if Γ has finite covolume, then the profinite completions of H and Γ are not isomorphic. If H has finite index in Γ, then there is a finite group onto which H maps but Γ does not. These results streamline the existing proofs that there exist full-sized groups that are profinitely rigid in the absolute sense. They build on a circle of ideas that can be used to distinguish among the profinite completions of subgroups of finite index in other contexts, for example, limit groups. We construct new examples of profinitely rigid groups, including the fundamental group of the hyperbolic 3-manifold Vol(3) and that of the 4-fold cyclic branched cover of the figure-eight knot. We also prove that if a lattice in PSL(2,C) is profinitely rigid, then so is its normalizer in PSL(2,C).

Quantum invariants of hyperbolic knots and extreme values of trigonometric products

In this paper, we study the relation between the function J_{4_1,0}, which arises from a quantum invariant of the figure-eight knot, and Sudler’s trigonometric product. We find J_{4_1,0} up to a constant factor along continued fraction convergents to a quadratic irrational, and we show that its asymptotics deviates from the universal limiting behavior that has been found by Bettin and Drappeau in the case of large partial quotients. We relate the value of J_{4_1,0} to that of Sudler’s trigonometric product, and establish asymptotic upper and lower bounds for such Sudler products in response to a question of Lubinsky.

Classifying the expanding attractors on the figure‐eight knot exterior and the non‐transitive Anosov flows on the Franks–Williams manifold

The figure-eight knot exterior N 0 $N_0$ supports a natural DA (derived from Anosov) expanding attractor, with which Franks–Williams constructed the first example of non-transitive Anosov flow. This flow lies in a 3-manifold M 0 $M_0$ which is the double of N 0 $N_0$ . We call M 0 $M_0$ by the Franks–Williams manifold. In this paper, we prove that, up to orbit-equivalence, this DA expanding attractor is the unique expanding attractor supported by N 0 $N_0$ . We also show that, up to orbit-equivalence, the non-transitive Anosov flow constructed by Franks and Williams is the unique non-transitive Anosov flow supported by M 0 $M_0$ . We also extend these results to a more general context.

Infinitesimal gluing equations and the adjoint hyperbolic Reidemeister torsion

We establish a link between the derivatives of Thurston's hyperbolic gluing equations on an ideally triangulated finite volume hyperbolic 3-manifold and the cohomology of the sheaf of infinitesimal isometries. This provides a geometric reformulation of the non-abelian Reidemeister torsion corresponding to the adjoint of the monodromy representation of the hyperbolic structure. These results are then applied to the study of the `1-loop Conjecture' of Dimofte--Garoufalidis, which we generalize to arbitrary 1-cusped hyperbolic 3-manifolds. We verify the generalized conjecture in the case of the sister manifold of the figure-eight knot complement.

A two-variable series for knot complements

The physical 3d \mathcal N = 2 theory T[Y] was previously used to predict the existence of some 3 -manifold invariants \widehat{Z}_{a}(q) that take the form of power series with integer coefficients, converging in the unit disk. Their radial limits at the roots of unity should recover the Witten–Reshetikhin–Turaev invariants. In this paper we discuss how, for complements of knots in S^3 , the analogue of the invariants \widehat{Z}_{a}(q) should be a two-variable series F_K(x,q) obtained by parametric resurgence from the asymptotic expansion of the colored Jones polynomial. The terms in this series should satisfy a recurrence given by the quantum A-polynomial. Furthermore, there is a formula that relates F_K(x,q) to the invariants \widehat{Z}_{a}(q) for Dehn surgeries on the knot. We provide explicit calculations of F_K(x,q) in the case of knots given by negative definite plumbings with an unframed vertex, such as torus knots. We also find numerically the first terms in the series for the figure-eight knot, up to any desired order, and use this to understand \widehat{Z}_a(q) for some hyperbolic 3-manifolds.