arXiv research

Proves uniqueness of holomorphic quilts on surfaces.

Study shows how tangle geometry maps onto pillowcase surfaces.

Detects figure-eight knot using Khovanov homology.

Study shows $(2,1)$-cable of figure-eight knot can't be smoothly sliced.

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.

New constructions show stable geodesics and figure-eights in convex hypersurfaces.

Proved colored HOMFLY-PT polynomials for specific 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 $S^3$ 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 $4s$.

Proves volume conjectures for figure-eight knot surgeries.

Study of lambda lengths in figure eight knot complement using Eisenstein integers.

Verifies a conjecture for the 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…

Study on the growth of colored Jones polynomial for figure-eight knot cables.

The paper proves rigidity of surgeries on the figure-eight knot complement.

Authors prove quantum invariant conjecture for figure-eight knot complement.

Study shortest non-simple geodesics on 2-orbifolds, finding unique shortest curve.

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.

In this paper, we compute the symplectic Floer homology of the figure eight knot. This provides first nontrivial knot with trivial symplectic Floer homology.

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…

In this paper we find infinitely many lattices in $SL(4,\mathbb{R})$ each of which contains thin subgroups commensurable with the figure-eight knot group.

New example shows figure eight knot not smoothly concordant but homology cobordant.

Study on colored Jones polynomial of figure-eight knot for complex parameters.

Study shows quantum modularity in 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.

The colored Jones polynomial of the figure-eight knot connects to an SL(2;R) representation.

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…

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.

We show that all positive contact surgeries on every Legendrian figure-eight knot in $(S^3, ξ_{\rm{std}})$ result in an overtwisted contact structure. The proof uses convex surface theory and invariants from Heegaard Floer homology.

Study confirms a 2-sphere metric with three geodesics of minimal length.

Study calculates quantum hyperbolic invariants for figure-eight knot complement, finding it either 0 or half the volume.

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…

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 $ω(p/q)\leqslant 12$, 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.

We calculate limits of the colored Jones polynomials of the figure-eight knot and conclude that in most cases they determine the volumes and the Chern--Simons invariants of the three-manifolds obtained by Dehn surgeries along it.

Optimal inequalities for systole, inradius, and volume in hyperbolic 3-manifolds

Motivated by Legendrian curve shortening flows in $\mathbb{R}^{3}$, 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.

Two algorithms use normal surfaces to detect unknots and prove knots.

Study shows exponential growth of knot polynomial tied to Chern-Simons invariant.

Explains exceptional surgeries connecting maps and knot orbifolds.