Study on volume and determinant densities of hyperbolic rational links.
problem Properties and distributions of volume and determinant densities of hyperbolic rational links.
method Construction of sequences of alternating knots and analysis of density distributions.
result Volume and determinant densities of hyperbolic rational links converge to any value in the interval [0,v_{oct}].
Study shows most hyperbolic knot complements lack hidden symmetries.
problem Proving most knot complements lack hidden symmetries.
method Using rational functions on varieties associated to a link.
result Most knot complements lack hidden symmetries.
Identifies a mod-p triple cup product for rational homology 3-spheres with specific first homology.
problem Locally flat embeddings in S4 for rational homology 3-spheres method Using triple torsion linking form and torsion-linking duality
result Identifies the mod-p triple cup product for specific rational homology 3-spheres New findings on L-spaces and taut foliations in hyperbolic links.
problem Characterizing L-spaces and taut foliations in Dehn surgeries on hyperbolic links. method Analyzing rational homology spheres and using coorientable taut foliations.
result Non-meridional surgeries on fibered hyperbolic two-bridge links support coorientable taut foliations.
We identify 998 closed hyperbolic 3-manifolds whose volumes are rationally related to Dedekind zeta values, with coprime integers a and b giving a/bvol(M)=(−D)3/2/(2π)2n−4(ζK(2))/(2ζ(2)) for a manifold M whose invariant trace field K has a single complex place, discriminant D, degree n, and Dedekin…
In this paper, we study the hyperbolicity of arborescent tangles and arborescent links. We will explicitly determine all essential surfaces in arborescent tangle complements with non-negative Euler characteristic, and show that given an arborescent tangle T, the complement X(T) is non-hyperbolic if and only if T is a r…
Paper develops geometry for Kleinian groups using Farey polynomials.
problem Understanding the geometry of Kleinian groups generated by parabolic elements.
method Sakuma-Weeks triangulations and Farey recursive polynomials.
result Simple recursive algorithm to determine link complement geometry.
Solves tangle equations involving composite links under specific conditions.
problem Solving tangle equations with composite links and 2-bridge links.
method Uses algebraic and double branched cover properties to solve equations.
result Non-hyperbolic solutions to tangle equations involving composite links are found.
Paper proves vanishing homology groups for certain hyperbolic groups.
problem Understanding homology groups of specific hyperbolic groups.
method Using twisted Wirtinger presentations to prove homology group vanishing.
result Second homology groups vanish for certain Gromov hyperbolic groups.
It is known that every closed oriented 3-manifold is homology cobordant to a hyperbolic 3-manifold. By contrast we show that many homology cobordism classes contain no Seifert fibered 3-manifold. This is accomplished by determining the isomorphism type of the rational cohomology ring of all Seifert fibered 3-manifolds …
The study calculates the average genus of rational knots and links.
problem Finding the average genus of rational knots and links.
method Enumerating and calculating the number of rational knots and links with a given crossing number.
result A precise formula for the average minimal genus of rational knots and links.
The paper enhances representations to show left-orderability of certain 3-manifold groups.
problem Left-orderability of 3-manifold groups using enhanced representations.
method Recalibration of Calegari and Dunfield's flipping construction for $\mbox{Homeo}_+(S^1)$-representations.
result Branched covers of links are left-orderable, generalizing known results.
Rational knots and links in solid torus characterized by continued fractions.
problem Characterizing rational knots and links in solid torus.
method Using rational tangles and continued fractions, and generalizing to skein module invariants.
result Rational links in solid torus fully characterized by rational tangles and continued fractions.
Suppose M is a hyperbolic 3-manifold which admits two Dehn fillings M(r1) and M(r2) such that M(r1) contains an essential torus and M(r2) contains an essential annulus. It is known that Δ=Δ(r1,r2)≤5. We will show that if Δ=5 then M is the Whitehead sister link exterior, and if Δ=4 then …
The paper characterizes Conway-Coxeter friezes using rational links.
problem Characterizing Conway-Coxeter friezes of zigzag type.
method Characterization via rational links and application to Jones polynomial.
result Jones polynomial can be defined for Conway-Coxeter friezes of zigzag type.
The paper converts nonalternating forms of rational links into all-even forms and derives formulas for their braid index and HOMFLY polynomial.
problem Finding formulas for the braid index and HOMFLY polynomial of rational links.
method Algorithm to transform nonalternating forms into all-even forms and derivation of formulas.
result Formulas for the braid index and HOMFLY polynomial of rational links in terms of their reduced alternating form.
The paper computes lens spaces resulting from rational surgeries on Hopf links.
problem Understanding rational surgeries on Hopf links in 3-sphere.
method Calculus of continued fractions.
result Explicit computation of resulting lens spaces.
We give an explicit formula for the Jones polynomial of any rational link in terms of the denominators of the canonical continued fraction of the slope of the given rational link.
New formulas derived for Jones polynomial of rational links.
problem Calculating the Jones polynomial of rational links.
method Colored Brylawski's tensor product formula for Tutte polynomials, finite automaton for crossing signs.
result Generalization of existing formulas for rational links.
Classifies L-space surgeries on all two-bridge links
problem Classifying L-space surgeries on two-bridge links method Introduces a sufficient diagrammatic condition for links in S3 to be persistently foliar, defines a simplified model for Heegaard Floer homology, and uses Turaev torsions for computations result Determines L-space surgeries in the case of generalised L-space links We prove properties of linking forms on rational homology spheres.
problem Properties of linking forms on rational homology spheres.
method Use of Heegaard splittings and properties of Q/Z-valued linking forms. result Linking forms on rational homology spheres are symmetric or anti-symmetric.
Given a sub-hyperbolic semi-rational branched covering which is not CLH-equivalent a rational map, it must have the non-empty canonical Thurston obstruction. By using this canonical Thurston obstruction, we decompose this dynamical system in this paper into several sub-dynamical systems. Each of these sub-dynamical sys…
New geometric proof for rational tangles links-quivers correspondence.
problem Recovering symmetric/antisymmetric colored HOMFLY-PT polynomials from a quiver.
method Geometric approach using winding numbers in punctured plane and its second configuration space.
result Explicit description of quivers for rational tangles.
Study shows conditions for rational ellipticity of manifolds with symmetries.
problem Conditions for rational ellipticity of manifolds with symmetries.
method Analyzes conditions on compact simply connected manifolds with G-actions. result Proves rational ellipticity of M/G if M satisfies certain conditions. 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 give an explicit formula for the HOMFLY polynomial of a rational link (in particular, a knot) in terms of a special continued fraction for the rational number that defines the given link.
The paper calculates the number of oriented rational links with a given deficiency.
problem Counting oriented rational links with a specific deficiency.
method Derived precise formulas for the number of oriented rational links with crossing number n and deficiency d.
result Precise formulas for the number of oriented rational links with crossing number n and deficiency d.
Constructs chiral rational homology spheres with hyperbolic groups.
problem Existence of strongly chiral rational homology spheres with hyperbolic fundamental groups.
method Construction of rational homology spheres using r-spins and investigation of self-map degrees. result Strongly chiral rational homology spheres with hyperbolic fundamental groups constructed.
Geometrically describes the linear and quadratic forms for rational links.
problem Predicting generating functions for colored HOMFLY-PT polynomials of rational links.
method Direct geometric description of linear and quadratic forms in terms of configuration spaces.
result Direct geometric description of forms for rational links.
Study verifies asymptotic expansion for Reshetikhin-Turaev invariants of fundamental shadow link pairs.
problem Verifying the asymptotic expansion conjecture for Reshetikhin-Turaev invariants of fundamental shadow link pairs.
method Using logarithmic holonomies of meridians and hyperbolic cone structures, the study verifies the conjecture for pairs where M∖L is homeomorphic to a fundamental shadow link complement. result The asymptotic expansion conjecture is true for pairs (M,L) with sufficiently small cone angles and M∖L homeomorphic to a fundamental shadow link complement. A theory of signatures for odd-dimensional links in rational homology spheres is studied via their generalized Seifert surfaces. The jump functions of signatures are shown invariant under appropriately generalized concordance and a special care is given to accommodate 1-dimensional links with mutual linking. Furthermor…
Classifies links in 3-sphere using Whitney towers in rational homology 4-ball.
problem Classifying links in 3-sphere using geometric methods.
method Complete classifications of links using Whitney towers in rational homology 4-ball.
result Geometric characterization of Milnor invariants and higher order Arf invariants.
This paper gives new and elementary combinatorial topological proofs of the classification of unoriented and oriented rational knots and links. These proofs are based on the known classification of alternating knots through flyping, and the calculus of continued fractions. We characterize the class of strongly invertib…
Paper proves triple linking form vanishes under specific conditions.
problem Analyzing rational homology cobordism and linking forms.
method Proves vanishing of triple torsion linking form under specific conditions.
result Triple torsion linking form vanishes on a specific Lagrangian.
Study spectral gaps in hyperbolic rational homology spheres.
problem Finding spectral gaps in hyperbolic rational homology spheres.
method Construction of families of hyperbolic rational homology spheres with coexact 1-form spectral gaps.
result Provided intervals containing limit points of spectral gaps, with the rightmost interval being [0.8196, 0.8277].
This paper gives two new combinatorial topological proofs of the classification of rational tangles. Each proof rests on an elegant lemma showing that rational tangles are isotopic to canonical alternating rational tangles. The first proof defines the tangle fraction from the canonical form and uses flyping to prove in…
New theorem allows transverse links to be braided with rational book structure.
problem Transverse isotopy of links in contact 3-manifolds.
method Generalization of Pavelescu's argument for rational open book decomposition.
result Every transverse link can be isotoped to a braid with rational open book.
Paper finds linking numbers for Montesinos links using a simple algorithm.
problem Calculating linking numbers for Montesinos links.
method Simple proof and numerical algorithm for rational links, extending to Montesinos links.
result Linking numbers found for any two components in Montesinos links.
Only finitely many rational multiples of π are angles between geodesics on hyperbolic surfaces.
problem Characterizing angles between geodesics on hyperbolic surfaces.
method Analyzing the set of angles between closed geodesics on hyperbolic surfaces of finite type.
result There are only finitely many rational multiples of π in the set of angles between geodesics.
Classifies fertility of all rational links.
problem Understanding the resultant and fertility of knots and links.
method Introduced the concept of link fertility and classified rational links.
result All rational links have a fertility number.
Rationally null-homologous links in Seifert fibered spaces may be represented combinatorially via labeled diagrams. We introduce an additional condition on a labeled link diagram and prove that it is equivalent to the existence of a rational Seifert surface for the link. In the case when this condition is satisfied, we…
Determines surgeries on chain links bounding rational homology balls using lattice-theoretic methods.
problem Integral surgeries on chain links bounding rational homology balls.
method Lattice-theoretic cubiquity obstruction and practical computation methods.
result Proves slice-ribbon conjecture for quasi-alternating 3-braid links, extending previous results.
We construct examples of hyperbolic rational homology spheres and hyperbolic knot complements in rational homology spheres containing closed embedded totally geodesic surfaces.
An L-space is a rational homology 3-sphere with minimal Heegaard Floer homology. We give the first examples of hyperbolic L-spaces with no symmetries. In particular, unlike all previously known L-spaces, these manifolds are not double branched covers of links in S^3. We prove the existence of infinitely many such examp…
This paper explores how rational numbers on the Stern-Brocot diagram map to lines when terms are extended.
problem Understanding the geometry of rational numbers on the Stern-Brocot diagram.
method Analyzing continued fraction expansions and their geometric implications on the diagram.
result Vertices of the Stern-Brocot diagram corresponding to extended rational numbers lie on two Euclidean lines.
Alexander polynomial linked to twist sites in rational links.
problem Relating Alexander polynomial to twist sites in rational links.
method Using Kauffman's clock moves and a lattice for Alexander polynomial terms.
result Alexander polynomial value at (-1,0) determined by twist sites.
Paper finds new 3D shapes that can be inside a 4D space.
problem Finding new 3D shapes with specific properties.
method Examined arithmetic hyperbolic 3-manifolds and their homology.
result Discovered infinitely many 3D shapes that are rational homology spheres and can bound geometrically.
Paper connects Conway-Coxeter friezes and rational tangles.
problem Understanding the relationship between Conway-Coxeter friezes and rational tangles.
method Using Kauffman bracket polynomials to compute and connect friezes and rational tangles.
result Provides a complete invariant for Conway-Coxeter friezes of zigzag-type.