Study lattice paths from twist knots and double twist knots.
problem Understanding combinatorics of twist knots and double twist knots.
method Analyzing quiver generating series of HOMFLY-PT polynomial limits.
result Lattice path models for twist knots and double twist knots.
The paper introduces colorings and invariants for twisted links and shows how double coverings can be equivalent.
problem Understanding and distinguishing twisted links.
method Twisted intersection colorings and double coverings.
result There exist infinitely many pairs of twisted links with equivalent double coverings.
Proved cyclotomic expansion for double twist knots' HOMFLY-PT invariants.
problem Proving cyclotomic expansion for colored HOMFLY-PT invariants of double twist knots.
method Cyclotomic expansion formula for double twist knots.
result Confirmed cyclotomic expansion conjecture for SU(N)-invariants.
Twisted links are a generalization of virtual links. As virtual links correspond to abstract links on orientable surfaces, twisted links correspond to abstract links on (possibly non-orientable) surfaces. In this paper, we introduce the notion of the double covering of a twisted link. It is defined by considering the o…
Formulae for volumes of double twist knots found.
problem Calculating volumes of specific hyperbolic cone-manifolds.
method Explicit formulae for volumes of double twist knots.
result Formulae for volumes of double twist knots found.
Computes Jones polynomial for specific knots.
problem Computing Jones polynomial for double twist knots.
method Using cyclotomic expansion and Kauffman bracket skein theory.
result Answers a question about Jones polynomial for specific knots.
The paper constructs triangulations for double twist knots using geometric methods.
problem Constructing explicit triangulations of double twist knots.
method Using triangulating Dehn fillings, layered solid tori, and their double covers.
result Proves both triangulations are geometric, using conjecturally minimal triangulation to present A-polynomial equations.
The paper finds left orderable slopes for specific double twist knots.
problem Identifying left orderable slopes for double twist knots.
method Analyzing the fundamental group of 3-manifolds obtained by surgery on double twist knots.
result Explicit intervals of rational numbers are left orderable slopes for certain double twist knots.
The paper proves that rational concordance of double twist knots is reciprocal.
problem Determining when double twist knots are rationally slice.
method Using Donaldson's diagonalization theorem, the paper constructs rational ball obstructions for non-rational sliceness.
result Infinitely many prime power-fold cyclic branched covers do not bound a rational ball, proving the converse of rational concordance.
Extends double symplectic groupoids to transitive Courant algebroids.
problem Generalizing double symplectic groupoids to a broader class of Lie bialgebroids.
method Classification of exact twisted Courant algebroids and construction of foliations.
result Generalization of many examples of double symplectic groupoids.
Generalizes Hecke algebra for double torus, linking to skein algebra.
problem Understanding algebraic structures on double torus.
method Introducing Heegaard dual operators and Dehn twists.
result Established relationship between Hecke algebra and skein algebra.
Proves volume conjecture for double twist knots using complexified tetrahedrons.
problem Volume conjecture for double twist knots
method Complexified tetrahedron and associated SL(2, C) representation of fundamental group
result Volume conjecture proved for double twist knots
The paper generalizes Kuperberg invariants using twisted Drinfeld doubles.
problem Quantum invariants of knots with additional structure.
method Using twisted Drinfeld doubles and Reshetikhin-Turaev invariants.
result Reidemeister torsion of knot complements as a quantum invariant.
Study on nonorientable 4-genus of double twist knots.
problem Determining the nonorientable 4-genus of double twist knots.
method Explicit constructions and obstructions from Donaldson's diagonalization theorem.
result Proved bounds on nonorientable 4-genus for infinite subfamilies of double twist knots.
A conjecture of Riley about the relationship between real parabolic representations and signatures of two-bridge knots is verified for double twist knots.
New formulas for colored Jones polynomials of double twist knots and related series.
problem Calculating colored Jones polynomials and related series for double twist knots.
method Utilized Takata's result and Bailey pairs, along with Walsh's formulas.
result Found new families of q-hypergeometric series generalizing the Kontsevich-Zagier series. New method untangles double-twist loops efficiently.
problem Tangling of double-twist loops in 3D space.
method Geometrically defined waving procedure to untangle loops.
result Minimum complexity of untanglings is reduced.
In this paper we present a certain class of geodesic vector fields of the double-twisted product R X R. Some examples of totally geodesic foliations are given.
Extends Drinfel'd doubles to include twists and generalizes Lie bialgebroids.
problem Tackles the extension of Drinfel'd doubles and Lie bialgebroids to include twists and generalizes them.
method Introduces a framework of calculus on algebroids and examines compatibility conditions for various algebroid properties.
result Introduces the notion of proto bialgebroids and their Drinfel'd doubles, generalizing both bialgebroids and proto Lie bialgebroids.
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.
New formulas for colored Jones polynomials of double twist knots generalize series and duality.
problem Calculating colored Jones polynomials for double twist knots.
method Using Takata's result and comparing with cyclotomic expansions.
result Generalizes Kontsevich-Zagier series and duality at roots of unity.
Paper discusses a new link invariant from virtual link theory.
problem Developing a new invariant for twisted links.
method Using JKSS invariant of virtual links and double coverings.
result Derived a new invariant for twisted links.
The paper proves properties of branched covers of specific knots and tori.
problem Investigating the smoothness and diffeomorphism of specific 4-manifolds.
method Analyzing double branched covers of twist-roll spun knots and turned twisted tori, applying techniques to show diffeomorphism.
result Proves that certain 4-manifolds are diffeomorphic to standard manifolds.
A new double quasi-Poisson bracket on surface groups.
problem Constructing a new mathematical structure on surface groups.
method Proposing and proving a double quasi-Poisson bracket on group algebras.
result The double quasi-Poisson bracket is a noncommutative generalization of the Goldman bracket.
The study finds generalized torsion elements in 3-manifolds from specific knot surgeries.
problem Identifying generalized torsion elements in 3-manifolds from knot surgeries.
method Using the JSJ-decomposition of the 3-manifold and bi-orderability properties.
result Existence and properties of generalized torsion elements in specific 3-manifolds.
Study determines left-orderable properties of knot covers.
problem Determining left-orderability of knot covers.
method Analyzing 2-bridge knots, including double-twist knots, using cyclic branched covers.
result Identifies specific integers for left-orderability of knot covers.
A double pants decomposition of a 2-dimensional surface is a collection of two pants decomposition of this surface introduced in arXiv:1005.0073v2. There are two natural operations acting on double pants decompositions: flips and handle twists. It is shown in arXiv:1005.0073v2 that the groupoid generated by flips and h…
We consider a homology sphere Mn(K1,K2) presented by two knots K1,K2 with linking number 1 and framing (0,n). We call the manifold {\it Matsumoto's manifold}. We show that there exists no contractible bound of Mn(T2,3,K2) if n<2τ(K2) holds. We also give a formula of Ozsváth-Szabó's τ-invariant as…
Double pants decompositions were introduced in our paper "Double pants decompositions of 2-surfaces" (Mosc. Math. J. 11 (2011), no. 2, 231-258, arXiv:1005.0073), together with a flip-twist groupoid acting on these decompositions. It was shown that flip-twist groupoid acts transitively on a certain topological class of …
New examples show high twisting doesn't guarantee open book maximality.
problem Maximality of open book pages under high twisting.
method Double branched covers of closed braids.
result Arbitrarily high twisting does not guarantee maximal Euler characteristic.
New examples of Schoenflies balls are produced using a 5D approach.
problem Identifying Schoenflies balls that are not standard.
method Using a 5-dimensional perspective, algebraic and geometric handle cancellation.
result New examples of Schoenflies balls not known to be standard are produced.
New algebra for twice-punctured torus curves.
problem Constructing a new algebra for skein theory.
method Using Heegaard dual of Iwahori--Hecke operator, Dehn twists are represented.
result Automorphisms correspond to Dehn twists on the twice-punctured torus.
Formulae for volumes and A-polynomials of a specific link.
problem Calculating volumes and A-polynomials for a specific type of link.
method Analyzing the double twist link and using character varieties.
result Formulas for the volume and A-polynomial of J(2m+1,2m+1). New method extracts Racah matrices from antiparallel double-braid knots.
problem Extract exclusive Racah matrices for arborescent knots.
method Factorization of differential expansion for antiparallel double-braids.
result Found a way to reduce problem to twist knots and provide answers for R=[33]. Researchers compute Chern-Simons invariants for a specific type of knot orbifolds.
problem Calculating Chern-Simons invariants for hyperbolic knot orbifolds.
method Used Schläfli formula for generalized Chern-Simons function on cone-manifold structures.
result Explicit formulae for the invariants of hyperbolic J(2n,−2m) knot orbifolds were derived. In this paper we study the knot Floer homology invariants of the twisted and untwisted Whitehead doubles of an arbitrary knot K. We present a formula for the filtered chain homotopy type of HFK(D(+,K,t)) in terms of the invariants for K, where D(+,K,t) denotes the t-twisted positive-clasped Whitehead double of K. In pa…
The study confirms a conjecture for a specific type of knot.
problem Determining the topology of essential surfaces in knot theory.
method Analyzing twisted, generalized Whitehead doubles of knots.
result Twisted, generalized Whitehead doubles satisfy the Slope and Strong Slope Conjectures.
We give an extension of Fox's formula of the Alexander polynomial for double branched covers over the three-sphere. Our formula provides the Reidemeister torsion of a double branched cover along a knot for a non-trivial one dimensional representation by the product of two factors derived from the knot group. One of the…
Study calculates non-trivial link cohomologies and applies to branched double covers.
problem Calculating non-trivial link cohomologies.
method Uses Baldwin-Ozsváth-Szabó cohomology and Heegaard-Floer homology.
result Demonstrates applications of cohomology to branched double covers.
Gluck twisting certain knots results in standard 4-spheres.
problem Understanding when Gluck twists yield standard 4-spheres.
method Analyzing smooth homotopy 4-spheres and diffeomorphisms.
result Infinite collection of twisted doubles of corks are standard.
DAHA and skein algebra on a double-torus knot are studied.
problem Understanding the relationship between DAHA and skein algebra on a double-torus surface.
method Combining DAHA of A1-type and CC1-type with the skein algebra on a 4-punctured sphere, constructing DAHA representation for a genus-two surface, and proposing a DAHA polynomial for a double-torus knot.
result A DAHA polynomial for a double-torus knot is proposed, and its relationship with the colored Jones polynomial is discussed.
Paper investigates double coverings and torsions in homology groups.
problem Relationship between cohomology ranks and twisted cohomology groups.
method Analyzes double coverings and torsions in homology groups of Milnor fiber.
result Strict version of Papadima-Suciu's inequality and non-trivial 2-torsion in homology are equivalent.
We use bordered Heegaard Floer homology to compute the tau invariant of a family of satellite knots obtained via twisted infection along two components of the Borromean rings, a generalization of Whitehead doubling. We show that tau of the resulting knot depends only on the two twisting parameters and the values of tau…
The tetrus is a sort of big brother to the tripus, W.P. Thurston's example of a compact hyperbolic 3-manifold with totally geodesic boundary. We describe a sixfold cover of the double of the tetrus, itself a double, which fibers over the circle with fiber a closed surface of genus 19. We also record arithmeticity of th…
Study on Whitehead doubles and their sliceness properties.
problem Understanding sliceness of Whitehead doubles of knots.
method Survey of techniques to obstruct sliceness and improve bounds on non-orientable genus.
result Improved bounds on non-orientable 4 genus of Whitehead doubles and genus 1 non-orientable cobordisms to cable knots.
We study twisted Jacobi manifolds, a concept that we had introduced in a previous Note. Twisted Jacobi manifolds can be characterized using twisted Dirac-Jacobi, which are sub-bundles of Courant-Jacobi algebroids. We show that each twisted Jacobi manifold has an associated Lie algebroid with a 1-cocycle. We introduce t…
In recent years, several families of hyperbolic knots have been shown to have both volume and λ1 (first eigenvalue of the Laplacian) bounded in terms of the twist number of a diagram, while other families of knots have volume bounded by a generalized twist number. We show that for general knots, neither the twist nu…
Paper studies convergence of Yang-Mills-Higgs flow on Kähler manifolds.
problem Analyzing convergence of Yang-Mills-Higgs flow for twisted Higgs pairs.
method Proves convergence to a reflexive twisted Higgs sheaf outside a closed subset.
result Limiting twisted Higgs sheaf is isomorphic to the double dual of graded twisted Higgs sheaves.