Explicitly constructs complex maps with lemniscate knot nodal lines.
problem Creating knotted fields with lemniscate knot nodal lines.
method Explicit construction of complex maps with lemniscate knots as nodal lines.
result Existence and fibrational properties of lemniscate knot nodal lines.
Knots in Euclidean space which may be parameterized by a single cosine function in each coordinate are called Lissajous knots. We show that twist knots are Lissajous knots if and only if their Arf invariants are zero. We further prove that all 2-bridge knots and all (3,q)-torus knots have Lissajous projections.
Knots can be transformed into Fourier knots using Lissajous knots.
problem Representing knots using Fourier series.
method Deformation of Lissajous knots to Fourier knots.
result Any knot is isotopic to a Fourier knot of type (1,1,2).
A Lissajous knot is one that can be parameterized by a single cosine function in each coordinate. Lissajous knots are highly symmetric, and for this reason, not all knots are Lissajous. We prove several theorems which allow us to place bounds on the number of Lissajous knot types with given frequencies and to efficient…
Classifies 3-braids from choreographic motions on Lissajous curves, linking them to mapping classes and geodesics.
problem Classifying 3-braids from choreographic motions on Lissajous curves.
method Parametrization in terms of levels and slopes, using dilatation and geodesic cutting sequences.
result Dilatation of pseudo-Anosov mapping classes increases with level or slope.
New Lissajous-toric knots studied with braid representations.
problem Characterizing and understanding Lissajous-toric knots.
method Investigates braid representations and properties of Lissajous-toric knots.
result Upper bounds for 4-genus and examples of trivial knots.
Knots parametrized in cylinder coordinates by t -> (st, 3 + cos(nt), cos(mt + φ)) share properties of Lissajous and billiard knots in a cylinder. We use these 'billiard knots in a flat solid torus' to study two topics: when is Z(s,n,m) equal to Z(s,m,n)? And: why are the determinants of certain Lissajous and billiard k…
Symmetry of geometrical figures is reflected in regularities of their algebraic invariants. Algebraic regularities are often preserved when the geometrical figure is topologically deformed. The most natural, intuitively simple but mathematically complicated, topological objects are Knots. We present in this papers seve…
We show that the Conway polynomials of Fibonacci links are Fibonacci polynomials modulo 2. We deduce that, when $ n \not\equiv 0 \Mod 4$ and (n,j)=(3,3), the Fibonacci knot $ \cF_j^{(n)} $ is not a Lissajous knot.
A Chebyshev knot is a knot which admits a parametrization of the form x(t)=Ta(t); y(t)=Tb(t); z(t)=Tc(t+φ), where a,b,c are pairwise coprime, Tn(t) is the Chebyshev polynomial of degree n, and $φ\in \RR .$ Chebyshev knots are non compact analogues of the classical Lissajous knots. We show that the…
Paper compares PCA of neural network training to high-dimensional random walks.
problem Understanding the dynamics of neural network training through PCA.
method PCA of neural network parameters and random walks, comparison of variances and projections.
result Most variance in neural network training and high-dimensional random walks is captured by the first few PCA components.
Detects figure-eight knot using Khovanov homology.
problem Detecting the figure-eight knot.
method Using Dowlin's spectral sequence from Khovanov homology to knot Floer homology.
result Reduced Khovanov homology (over Q) detects the figure-eight knot.
Study shows (2,1)-cable of figure-eight knot can't be smoothly sliced.
problem Determining if a knot can be smoothly sliced.
method Showed that the branched double cover of the (2,1)-cable of the figure-eight knot bounds no equivariant homology ball. result The (2,1)-cable of the figure-eight knot is not smoothly slice. Contact surgeries on figure-eight knots yield overtwisted structures.
problem Characterizing contact structures after surgeries on Legendrian knots.
method Convex surface theory and Heegaard Floer homology.
result All positive surgeries on figure-eight knots produce overtwisted contact structures.
The paper contains the computation of the noncommutative A-ideal of the figure-eight knot, a noncommutative generalization of the A-polynomial. We show that if a knot has the same noncommutative A-ideal as the figure-eight knot, then all colored Kauffman brackets are the same as those of the figure-eight knot.
Curve shortening flow on figure-eight curves in the plane.
problem Understanding curve behavior under shortening flow.
method Curve shortening flow applied to figure-eight curves.
result Figure-eight curves shrink to a point at first singular time under certain conditions.
Proves volume conjectures for figure-eight knot surgeries.
problem Volume conjectures for hyperbolic 3-manifolds.
method Ohtsuki's method applied to figure-eight knot surgeries.
result Proves Asymptotic Expansion and Volume Conjectures for figure-eight knot surgeries.
Smooth figure-eight knot cables have infinite order.
problem Proving infinite order of figure-eight knot cables.
method Introduced new concordance invariants via branched covers and real Seiberg-Witten Floer K-theory.
result Uniform proof for all (2n,1)-cables of the figure-eight knot. Explicit equations for SL(3,C) character variety of figure eight knot.
problem Describing the SL(3,C) character variety of the figure eight knot.
method Provided explicit equations and analyzed the five components of the character variety.
result Found three components of irreducible representations and distinguished one curve.
The paper finds infinitely many ways to triangulate the figure eight knot complement.
problem Finding infinitely many geometric triangulations of a specific 3-manifold.
method Examining ideal triangulations of cusped hyperbolic 3-manifolds, focusing on positive volume ideal hyperbolic tetrahedra.
result Constructs infinitely many geometric ideal triangulations of the figure eight knot complement.
New constructions show stable geodesics and figure-eights in convex hypersurfaces.
problem Constructing stable geodesics and figure-eights in convex hypersurfaces.
method Explicit billiard trajectories with controlled parallel transport in convex polytopes.
result Construction of stable figure-eights and index-zero geodesics in convex hypersurfaces.
Proved colored HOMFLY-PT polynomials for specific knots.
problem Calculating colored HOMFLY-PT polynomials for specific knots.
method Rigorous mathematical proof for trefoil, figure-eight, and twist knots.
result Colored HOMFLY-PT polynomials expressed as sums for different knots.
In this paper we show that some open set of the representations of the fundamental group of figure-eight knot complement found in \cite{Ballas12a} are the holonomies of a family of finite volume properly convex projective structures on the figure-eight knot complement.
Researchers conjecture HOMFLY polynomial for figure eight knot.
problem Finding HOMFLY polynomial for figure eight knot.
method Differential expansion for Wilson loop averages, focusing on rectangular representations.
result Conjecture for rectangularly colored HOMFLY polynomial of figure eight knot.
We show that most cabled knots over the figure eight knot in S3 satisfy the AJ-conjecture, in particular, any (r,s)-cabled knot over the figure eight knot satisfies the AJ-conjecture if r is not a number between −4s and 4s.
Study on knot polynomial's asymptotic behavior for figure eight.
problem Investigating the asymptotic behavior of colored HOMFLY polynomial for figure eight knot.
method Establishing an asymptotic expansion for the colored HOMFLY polynomial.
result Showed that Chern-Simons invariants and twisted Reidemeister torsion can be derived from the polynomial.
Study tight contact structures on figure-eight knot surgeries.
problem Classify tight contact structures on surgeries of figure-eight knot.
method Analyzes surgeries on figure-eight knot, determining tightness, symplectic fillability, and universality.
result First classification of tight contact structures on surgeries of figure-eight knot.
Study on asymptotic behavior of knot invariants for figure eight knot.
problem Investigate asymptotic behavior of colored Jones polynomials and Turaev-Viro invariants for figure eight knot.
method Considered M-th colored Jones polynomials and Turaev-Viro invariants for figure eight knot with fixed limiting ratio s of M and (N+1/2). Found asymptotic expansion formula for colored Jones polynomials and showed exponential growth rate difference for s close to 1/2 and 1. Related Turaev-Viro invariants to colored Jones polynomials. result Asymptotic expansion formula for colored Jones polynomials and Turaev-Viro invariants of figure eight knot.
Study of lambda lengths in figure eight knot complement using Eisenstein integers.
problem Determining lambda lengths in the figure eight knot complement.
method Using hyperbolic geometry and spinors, mapping lambda lengths to Eisenstein integers.
result Lambda lengths are precisely the Eisenstein integers, up to multiplication by a unit.
The figure-eight knot's complement bounds a hyperbolic 4-manifold.
problem Bounding hyperbolic knot complements in higher dimensions.
method Embedding tessellated 3-manifolds in 4-manifolds.
result The figure-eight knot's complement geometrically bounds a hyperbolic 4-manifold.
Verifies a conjecture for the figure eight knot.
problem Relates A-ideal and recurrence ideal of knots.
method Uses quantum A-ideals, q-holonomicity, and AJ conjecture.
result Strong AJ conjecture verified for figure eight knot.
We obtain a branched spherical CR structure on the complement of the figure eight knot with a given holonomy representation (called rho_2). There are essentially two boundary unipotent representations from the complement of the figure eight knot into PU(2,1), we call them rho_1 and rho_2. We make explicit some fundamen…
The paper proves rigidity of surgeries on the figure-eight knot complement.
problem Infinitesimal projective rigidity of surgeries on the figure-eight knot complement.
method Computer-assisted proof and explicit representations of the knot complement.
result Proves infinitesimal projective rigidity for surgeries far from the ideal point.
Study on the growth of colored Jones polynomial for figure-eight knot cables.
problem Asymptotic behavior of colored Jones polynomial for figure-eight knot cables.
method Analyzing the asymptotic growth of the N-dimensional colored Jones polynomial of a cable of the figure-eight knot. result The growth rate of the colored Jones polynomial is exponential and related to the Chern-Simons invariant.
Authors prove quantum invariant conjecture for figure-eight knot complement.
problem Connecting quantum invariants of surface diffeomorphisms to hyperbolic volumes.
method Analyzes the simplest case of a one-puncture torus and figure-eight knot complement.
result Proves conjecture linking quantum invariant to hyperbolic volume.
Study knot diagrams on a sphere without vertical lines, focusing on minimal crossings.
problem Understanding minimal crossings of knot diagrams on a punctured sphere.
method Mathematical model of string figures using knot diagrams on xyz-space with missing vertical lines, analyzing minimal crossings under Reidemeister moves. result Minimal number of crossings of knot diagrams on a punctured sphere.
Proves surfaces with specific intersections are cylinders.
problem Characterizing surfaces with clean figure-8 intersections.
method Analyzes intersections with planes and uses central symmetry.
result Proves surfaces with clean figure-8 intersections are cylinders.
Study shortest non-simple geodesics on 2-orbifolds, finding unique shortest curve.
problem Finding shortest non-simple closed geodesics disjoint from orbifold points.
method Fundamental domains and hyperbolic trigonometry.
result Identified and classified all figure eight geodesics on triangle group orbifolds.
Jones slopes detect figure eight knot, and characterize alternating knots.
problem Detecting knots using Jones polynomials.
method Strong slope conjecture and colored Jones polynomials.
result Jones slopes detect figure eight knot and characterize alternating knots.
We describe a simple fundamental domain for the holonomy group of the boundary unipotent spherical CR uniformization of the figure eight knot complement, and deduce that small deformations of that holonomy group (such that the boundary holonomy remains parabolic) also give a uniformization of the figure eight knot comp…
3-manifolds transform into figure-eight knot complements through complex hyperbolic deformations.
problem Understanding transformations of 3-manifolds into figure-eight knot complements.
method Deforming representations of complex hyperbolic triangle groups.
result The quotient space is always the figure-eight knot complement.
In this paper, we compute the symplectic Floer homology of the figure eight knot. This provides first nontrivial knot with trivial symplectic Floer homology.
New example shows figure eight knot not smoothly concordant but homology cobordant.
problem Smooth concordance vs homology cobordism of knots.
method Construction of knots with specific properties.
result Figure eight knot not smoothly concordant but homology cobordant.
Formula for Reidemeister torsion after knot surgery.
problem Calculating the Reidemeister torsion of a 3-manifold after surgery.
method Provided a formula for the torsion using the trace of the meridian image.
result Reidemeister torsion is described as a rational expression of the trace.
In this paper we find infinitely many lattices in SL(4,R) each of which contains thin subgroups commensurable with the figure-eight knot group.
Researchers compute Reidemeister torsion for a specific 3-sphere.
problem Computing Reidemeister torsion for a specific type of 3-manifold.
method Numerical computations on representations of fundamental group in SL(2;C) and Reidemeister torsion.
result Corrected and presented new computations of Reidemeister torsion.
Study on colored Jones polynomial of figure-eight knot for complex parameters.
problem Asymptotic behavior of colored Jones polynomial for figure-eight knot.
method Analyzing the asymptotic growth rate of the polynomial for complex parameters with small imaginary part.
result Growth rate of polynomial is related to the Chern-Simons invariant for large real part of the parameter and to the reciprocal of Alexander polynomial for small real part.
Study shows quantum modularity in figure-eight knot's colored Jones polynomial.
problem Asymptotic behavior of colored Jones polynomial of figure-eight knot.
method Analyzing polynomial evaluated at specific points and showing asymptotic equivalence.
result Quantum modularity demonstrated in the figure-eight knot's colored Jones polynomial.