Continuity of roots of hyperbolic polynomials with smooth coefficients.
problem Continuity of the solution map for hyperbolic polynomials.
method Proving continuity of the solution map from hyperbolic polynomials of degree d with C^d coefficients to their increasingly ordered roots.
result Continuity of the solution map for hyperbolic polynomials with C^d coefficients.
Finite quotients of fibered hyperbolic 3-manifold groups detect taut polynomials.
problem Detecting taut polynomials of fibered faces of Thurston norm balls
method Developing a framework for profinite invariance of twisted multivariable Alexander polynomials
result Proving finite quotients detect taut polynomials
Paper analyzes double twist knots using adjoint hyperbolic torsion polynomial.
problem Determining the genus and fibering of double twist knots.
method Uses adjoint hyperbolic torsion polynomial to analyze double twist knots.
result The adjoint hyperbolic torsion polynomial determines the genus and fibering of double twist knots.
Explicit polynomial bound found for subgroup Dehn function.
problem Finding explicit bounds on Dehn functions of subgroups of hyperbolic groups.
method Constructing a specific example of a non-hyperbolic subgroup and analyzing its Dehn function.
result Explicit polynomial upper bound n96 on the Dehn function of a non-hyperbolic subgroup. Polynomial bound on surfaces in hyperbolic 3-manifolds.
problem Bounding the number of surfaces in hyperbolic 3-manifolds.
method Using polynomial functions of the volume of the manifold and the Euler characteristic.
result An upper bound for the number of compact essential surfaces is a polynomial function of the volume of the manifold.
We realize a given (monic) Alexander polynomial by a (fibered) hyperbolic arborescent knot and link of any number of components, and by infinitely many such links of at least 4 components. As a consequence, a Mahler measure minimizing polynomial, if it exists, is realized as the Alexander polynomial of a fibered hyperb…
The paper connects knot volume to A-polynomial structure.
problem Understanding the relationship between knot volume and A-polynomial structure. method Examining satellite knots and their A-polynomials to conjecture a connection with hyperbolic volume. result The conjecture that knots with zero hyperbolic volume have A-polynomials with specific factor structure. In this paper we apply the twisted Alexander polynomial to study the fibering and genus detecting problems for oriented links. In particular we generalize a conjecture of Dunfield, Friedl and Jackson on the torsion polynomial of hyperbolic knots to hyperbolic links, and confirm it for an infinite family of hyperbolic 2…
The paper confirms Arnold's conjecture about hyperbolic polynomials.
problem The number of connected components of hyperbolic polynomials increases linearly with degree.
method Constructive proof using homotopy invariance of the index of a curve and properties of homogeneous polynomials.
result Exact number of connected components of Hyp(D) is determined and representatives for each component are provided. The study classifies polynomial relation tubular surfaces in 3-spaces.
problem Classifying tubular surfaces with polynomial curvature relations.
method Analyzing polynomial relations between Gaussian and mean curvatures in Euclidean, hyperbolic, and Lorentzian 3-spaces.
result Determination of sets of polynomial relations for tubular surfaces.
Eigenvalues of random hyperbolic surface covers converge to hyperbolic plane's.
problem Eigenvalue rigidity of random hyperbolic surface covers.
method Selberg trace formula and polynomial method.
result Distribution of eigenvalues converges to hyperbolic plane's spectral measure.
3-manifolds with similar completions have matching slopes and polynomials.
problem Matching slopes and polynomials in cusped hyperbolic 3-manifolds.
method Prove similarity of profinite completions leads to matching A-polynomials and boundary slopes. result Strongly detected boundary slopes match between manifolds with similar completions.
Proves regularity of isomorphisms between hyperbolic 3-manifolds.
problem Regularity of isomorphisms between cusped hyperbolic 3-manifolds.
method Proves regularity of profinite completions of fundamental groups.
result Proves A-polynomial is a profinite invariant. Random covers of hyperbolic surfaces have a spectral gap with polynomial rate.
problem Finding spectral gaps in random covers of hyperbolic surfaces.
method Applying recent work on spectral gaps to uniformly random covers of closed hyperbolic surfaces.
result Uniformly random degree-n covers of a closed hyperbolic surface have no new Laplacian eigenvalues below a specific threshold with high probability.
Maps and embeddings between hyperbolic spaces and their boundaries studied.
problem Understanding relations between maps and embeddings between relatively hyperbolic spaces and their boundaries.
method Establishing correspondences between quasi-isometric embeddings and quasisymmetric embeddings, using polynomial distortion.
result Characterization of hyperbolic relative groups with polynomial distortion embeddings.
Classifies special homogeneous surfaces with unique properties.
problem Classifying hyperbolic polynomials with specific actions.
method Analyzes hyperbolic level sets and restricts negative Hessians.
result Existence of a finite number of special homogeneous surfaces with unique curvature.
Researchers compute and predict knot volumes using colored Jones polynomials.
problem Computing and predicting volumes of hyperbolic knots.
method Vertex model approach, neural network training, polynomial evaluations.
result 3-colored Jones polynomials predict knot volumes with high accuracy.
For a hyperbolic knot and a natural number n, we consider the Alexander polynomial twisted by the n-th symmetric power of a lift of the holonomy. We establish the asymptotic behavior of these twisted Alexander polynomials evaluated at unit complex numbers, yielding the volume of the knot exterior. More generally, we pr…
The study classifies geometrically finite polynomials on the boundary of Blaschke products.
problem Understanding the boundaries of hyperbolic components of Blaschke products.
method Combinatorial classification and construction of self-bumps.
result The closure of the main hyperbolic component is not a topological manifold with boundary for d≥4. New knots with specific properties have identical polynomial values.
problem Identifying knots with matching polynomial values after braiding.
method Constructing infinitely many hyperbolic knots and analyzing their braided satellites.
result Mutually distinct hyperbolic knots have identical HOMFLY polynomial values up to given z-degrees. Kashaev limits of quantum A-polynomials reveal classical action vanishing and hyperbolic volume deformation.
problem Exploring the Kashaev limits of quantum A-polynomials. method Analyzing the double scaling quasiclassical limit.
result Identifying two phases in the Kashaev limit.
New knots share same Upsilon invariant despite different Alexander polynomials.
problem Identifying concordant knots via Upsilon invariant.
method Examined hyperbolic L-space knots and their Upsilon invariants.
result Infinitely many pairs of hyperbolic L-space knots with distinct Alexander polynomials share the same Upsilon invariant.
Empirical evidence suggests link polynomials can detect causality in spacetimes.
problem Detect causality in (2+1)-dimensional globally hyperbolic spacetimes. method Introduced a new invariant of certain tangles related to the Conway polynomial.
result The Conway polynomial does not detect causality in certain spacetime scenarios.
We show that given n>0, there exists a hyperbolic knot K with trivial Alexander polynomial, trivial finite type invariants of order <=n, and such that the volume of the complement of K is larger than n. This contrasts with the known statement that the volume of the complement of a hyperbolic alternating knot is bounded…
Groups with hyperbolic properties don't have strong Property (T).
problem Proving groups with hyperbolic properties don't have strong Property (T).
method Constructing an unbounded affine representation with polynomial growth.
result Groups with hyperbolic properties do not have strong Property (T).
We show examples of knots with the same polynomial invariants and hyperbolic volumes, with variously coinciding 2-cable polynomials and colored Jones polynomials, which are not mutants.
Homology growth of specific mapping tori vanishes for certain groups.
problem Homology growth of polynomially growing mapping tori in various groups.
method Proof of the cheap rebuilding property for specific groups.
result Torsion homology growth vanishes for Farber sequences in every degree.
We obtain bounds on hyperbolic volume for periodic links and Conway sums of alternating tangles. For links that are Conway sums we also bound the hyperbolic volume in terms of the coefficients of the Jones polynomial.
We give a refined upper bound for the hyperbolic volume of an alternating link in terms of the first three and the last three coefficients of its colored Jones polynomial.
The Volume conjecture claims that the hyperbolic Volume of a knot is determined by the colored Jones polynomial. The purpose of this article is to show a Volume-ish theorem for alternating knots in terms of the Jones polynomial, rather than the colored Jones polynomial: The ratio of the Volume and certain sums of coeff…
An important conjecture in knot theory relates the large-N, double scaling limit of the colored Jones polynomial JK,N(q) of a knot K to the hyperbolic volume of the knot complement, Vol(K). A less studied question is whether Vol(K) can be recovered directly from the original Jones polynomial …
Study on colored Jones polynomial and link complements.
problem Understanding the structure of link complements with arbitrary colors.
method Investigated the potential function of the colored Jones polynomial and established a relationship with hyperbolicity.
result Evidence supports the Chen-Yang conjecture on link complements.
We show that the Mahler measures of the Jones polynomial and of the colored Jones polynomials converge under twisting for any link. Moreover, almost all of the roots of these polynomials approach the unit circle under twisting. In terms of Mahler measure convergence, the Jones polynomial behaves like hyperbolic volume …
We present computational data and heuristic arguments which suggest that given a hyperbolic knot the volume correlates with its determinant, the Mahler measure of its Alexander polynomial and the Mahler measure of the twisted Alexander polynomial corresponding to the discrete and faithful SL(2,C)-representation.
New infinite family of hyperbolic L-space knots with specific semigroups.
problem Characterizing semigroups of L-space knots.
method Defined formal semigroups from Alexander polynomials and analyzed hyperbolic knots.
result Found an infinite family of hyperbolic L-space knots with semigroups generated by five elements.
HZ transform applied to knot polynomials reveals hyperbolic knot structures.
problem Understanding the structure of knot polynomials and their factorisability.
method Applying the Harer-Zagier transform to knot polynomials and character expansions.
result Construction of an infinite family of hyperbolic knots and proof of factorisability in the 3-strand case.
We identify all hyperbolic knots whose complements are in the census of orientable one-cusped hyperbolic manifolds with eight ideal tetrahedra. We also compute their Jones polynomials.
The Hessian Topology is a subject with interesting relations with some classical problems of analysis and geometry. In this article we prove a conjecture on this subject stated by V.I. Arnold concerning the number of connected components of hyperbolic homogeneous polynomials of degree n. The proof is constructive and…
Bounding twist number of surface links using polynomial coefficients.
problem Bounding the twist number of alternating surface links.
method Introducing a generalized homological Kauffman bracket and applying it to surface link diagrams.
result A bound for the twist number of alternating surface links in terms of polynomial coefficients.
Proves freely 2-periodic knots have two canonical components in their character variety.
problem Identifying freely 2-periodic knots in character varieties.
method Analyzes SL(2, C) character variety and hyperbolic torsion polynomial.
result Character variety has two canonical components for freely 2-periodic knots.
Study links in 3-manifolds, linking volume to polynomial coefficients.
problem Understanding the volume of hyperbolic links in 3-manifolds.
method Using Kauffman bracket functions and polynomial invariants, linking volume to polynomial coefficients.
result Coefficients of polynomial provide 2-sided linear bounds on the volume of hyperbolic links.
A new method computes Teichmüller polynomials from integer permutations.
problem Computing Teichmüller polynomials for fibered 3-manifolds.
method Using integer permutations to characterize pseudo-Anosov homeomorphisms and train tracks.
result Direct implementation of McMullen's algorithm for Teichmüller polynomials.
In this article, we classify the set of asymptotic mass-like invariants for asymptotically hyperbolic metrics. It turns out that the standard mass is just one example (but probably the most important one) among the two families of invariants we find. These invariants are attached to finite-dimensional representations o…
New polynomial connects knot genus to 3-manifold geometry.
problem Detecting knot genus from 3-manifold geometry.
method Ideal triangulation and super-Ptolemy assignments.
result New polynomial conjecturally agrees with torsion polynomial.
We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Ohtsuki intersect non-trivially. Moreover it is shown that the intersection is (at least includes) the set of Kashaev's quantum dilogarithm invariants for links. Therefore Kashaev's conjectu…
R.M. Kashaev conjectured that the asymptotic behavior of his link invariant, which equals the colored Jones polynomial evaluated at a root of unity, determines the hyperbolic volume of any hyperbolic link complement. We observe numerically that for knots 63, 89 and 820 and for the Whitehead link, the colored…
We study a twisted Alexander polynomial naturally associated to a hyperbolic knot in an integer homology 3-sphere via a lift of the holonomy representation to SL(2, C). It is an unambiguous symmetric Laurent polynomial whose coefficients lie in the trace field of the knot. It contains information about genus, fibering,…
Using the correspondence between Chern-Simons theories and Wess-Zumino-Witten models we present the necessary tools to calculate colored HOMFLY polynomials for hyperbolic knots. For two-bridge hyperbolic knots we derive the colored HOMFLY invariants in terms of crossing matrices of the underlying Wess-Zumino-Witten mod…