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.
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.
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.
Establishes criterion for smooth and contact characterizing slopes of Legendrian knots.
problem Relating smooth and contact characterizing slopes for Legendrian knots.
method Criterion based on uniqueness of isotopy class and classical invariants.
result Criterion for smooth characterizing slopes to be contact characterizing slopes.
The paper conjectures Khovanov homology can distinguish torus and twist knots.
problem Detecting and distinguishing knots using Khovanov homology.
method Examining all prime knots with up to 20 crossings, conjecturing Legendrian simplicity.
result Numerical evidence supports Khovanov homology distinguishing torus and twist knots.
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.
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.
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.
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.
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 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.
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.
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…
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.
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.
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.
New algebraic structures help distinguish Legendrian knots.
problem Distinguishing Legendrian knots from smooth knots.
method Defined Legendrian racks and used them to define invariants.
result Distinguished certain Legendrian knots.
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…
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.
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.
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.
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…
Legendrian products can be twist spuns under certain conditions.
problem Understanding when Legendrian products are twist spuns.
method Using Legendrian contact homology and augmentation variety analysis.
result Not all Legendrian products are twist spuns.
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.
We establish an upper bound ω(p/q) on the complexity of manifolds obtained by p/q-surgeries on the figure eight knot. It turns out that if ω(p/q)⩽12, the bound is sharp.
We will study the asymptotic behaviors of the colored Jones polynomials of the figure-eight knot. In particular we will show that for certain limits we obtain the volumes of the cone manifolds with singularities along the knot.