The curve shortening flow transforms figure-eight curves into bowties.
problem Transforming figure-eight curves into a specific shape under curve shortening flow.
method Applied curve shortening flow to figure-eight curves with specific properties, proving convergence to a quadrilateral.
result The renormalized limit of the flow converges to a quadrilateral called a bowtie.
Motivated by Legendrian curve shortening flows in R3, we study the curve shortening flow of figure-eight curves in the plane. We show that, under some symmetry and curvature conditions, a figure-eight curve will shrink to a point at the first singular time.
In this article we investigate the dynamics of special solutions to the surface diffusion flow of idealised ribbons. This equation reduces to studying the curve diffusion flow for the profile curve of the ribbon. We provide: (1) a complete classification of stationary solutions; (2) qualitative results on shrinkers, tr…
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.
We give explicit equations that describe the character variety of the figure eight knot for the groups SL(3,C), GL(3,C) and PGL(3,C). This has five components of dimension 2, one consisting of totally reducible representations, another one consisting of partially reducible representations, and three components of irred…
Three geometric analysis results on curve flows and Lie groups.
problem Analyzing geometric flows and Lie groups.
method Curve-shortening flow, point-wise curvature preserving flow, Lie group analysis.
result Interpolation between Sol and hyperbolic space in Lie groups.
Improved bounds on shortest geodesics with self-intersections on hyperbolic surfaces.
problem Quantifying the complexity of non-simple closed geodesics on hyperbolic surfaces.
method Analyzing the geometry of shortest figure eight curves and constructing geodesic representatives.
result Explicit upper bounds for the length of shortest geodesics with k self-intersections improved from 512 to 128. 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. 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.
Study of closed real plane curves with hyperelliptic genus three solutions.
problem Analyzing real plane curves with specific curvature equations.
method Examined real plane curves associated with the focusing gauged modified KdV equation of genus three.
result Showed closed real plane curves beyond Euler's figure-eight elastica.
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. 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.
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 character varieties of tangles to map immersed curves in the pillowcase.
problem Characterizing holonomy-perturbed traceless SU(2) character varieties.
method Examining marked tangles as endomorphisms in the cobordism category and using holonomy-perturbed traceless character variety functor.
result Endomorphisms of immersed curves in the pillowcase have the same image.
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.
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.
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.
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.
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.
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.
Study wave functions in complex Chern-Simons theory, finding integrality and rational points.
problem Understanding wave functions in complex Chern-Simons theory.
method Conjecture and prove integrality structure, develop techniques to determine wave functions at rational points.
result Wave functions have integrality structure and can be determined at rational points.
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.
This paper determines the minimal degree sequence for two compact rational knots, namely the trefoil and figure-eight knots. We find explicit projections with the minimal degree sequence of each knot. This is done by modifying a non-compact rational minimal-degree parameterization of the trefoil and figure-eight knots …
Study shows shortest periodic geodesic on hyperbolic orbisphere complements figure-eight knot.
problem Shortest periodic geodesic on hyperbolic orbisphere with cone points.
method Computation of linking numbers to show homeomorphism.
result Lift of shortest periodic geodesic is homeomorphic to figure-eight knot complement.
The colored Jones polynomial of the figure-eight knot connects to an SL(2;R) representation.
problem Asymptotic behavior of colored Jones polynomial for the figure-eight knot.
method Analyzing the polynomial's behavior as N approaches infinity and evaluating it at specific points.
result The polynomial corresponds to an SL(2;R) representation of the knot complement.
In this paper we investigate the asymptotic behavior of the colored Jones polynomials and the Turaev-Viro invariants for the figure eight knot. More precisely, we consider the M-th colored Jones polynomials evaluated at (N+1/2)-th root of unity with a fixed limiting ratio, s, of M and (N+1/2). We find out the…
Study on hyperbolic elastic flow, proving convergence and quantifying singularities.
problem Understanding singularities and convergence of elastic flow in hyperbolic plane.
method Analyzes closed and open curves with clamped boundary conditions, proving convergence without small energy assumption.
result Each singularity carries an energy cost of at least 8, and blow-ups are explicitly classified.
This paper considers "geometric" ideal triangulations of cusped hyperbolic 3-manifolds, i.e. decompositions into positive volume ideal hyperbolic tetrahedra. We exhibit infinitely many geometric ideal triangulations of the figure eight knot complement. As far as we know, this is the first construction of infinitely man…
Characterizes unknotted curves on Seifert surfaces of twist knots.
problem Identifying unknotted curves on Seifert surfaces of twist knots.
method Analyzing homologically essential simple closed curves on Seifert surfaces of genus one knots.
result Characterizes unknotted curves on Seifert surfaces of twist knots, including infinitely many for the figure eight knot and one for Whitehead doubles.
Study confirms a 2-sphere metric with three geodesics of minimal length.
problem Understanding the systolic, width, and Gromov-Guth metrics on a 2-sphere.
method Classical min-max and hyperbolic geometry tools.
result Figure-eight geodesics achieve the systolic, width, and Gromov-Guth metrics on a 2-sphere.
We show that all positive contact surgeries on every Legendrian figure-eight knot in (S3,ξstd) result in an overtwisted contact structure. The proof uses convex surface theory and invariants from Heegaard Floer homology.
Study calculates quantum hyperbolic invariants for figure-eight knot complement, finding it either 0 or half the volume.
problem Computing quantum hyperbolic invariants for knot complements.
method Computed the real part of the semi-classical limit of quantum hyperbolic invariants of the figure-eight knot complement.
result The real part is rigid and either 0 or half the hyperbolic volume of the knot complement.
Differential expansion (DE) for a Wilson loop average in representation R is built to respect degenerations of representations for small groups. At the same time it behaves nicely under some changes of the loop, e.g. of some knots in the case of 3d Chern-Simons theory. Especially simple is the relation between the …
We classify Legendrian torus knots and figure eight knots in the tight contact structure on the 3-sphere up to Legendrian isotopy. As a corollary to this we also obtain the classification of transversal torus knots and figure eight knots up to transversal isotopy.
We describe the explicit form and the hidden structure of the answer for the HOMFLY polynomial for the figure eight and some other 3-strand knots in representation [21]. This is the first result for non-torus knots beyond (anti)symmetric representations, and its evaluation is far more complicated. We provide a whole va…
The study examines complex tangles in Curve Shortening Flow singularities.
problem Classifying all knots in R3 is a challenging problem. method Examine solutions to plane Curve Shortening Flow to identify tangles.
result A vanishing n-loop converges to a 'squeezed bow-tie' under rescaling. We show that the figure eight knot complement admits a uniformizable spherical CR structure, i.e. it occurs as the manifold at infinity of a complex hyperbolic orbifold. The uniformization is unique provided we require the peripheral subgroups to have unipotent holonomy.