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.
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 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.
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 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.
A conjecture of Riley about the relationship between real parabolic representations and signatures of two-bridge knots is verified for double twist knots.
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.
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.
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.
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.
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.
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. 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.
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.
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.
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). 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. Study compares WRT and CGP invariants using Habiro's series.
problem Comparing WRT and CGP invariants for knots.
method Established relationship between Habiro's series and ADO invariants.
result Difference between WRT and CGP invariants determined by Habiro series.
This paper completes proofs for left orderable slopes of double twist knots.
problem Proving the left orderability of slopes for double twist knots.
method Continuous families of hyperbolic SL2(R) representations of knot groups.
result Any slope in (-4n, 4m) is a left orderable slope of C(2m+1, -2n).
We compute both natural and smooth models for the SL2(C) character varieties of the two component double twist links, an infinite family of two-bridge links indexed as J(k,l). For each J(k,l), the component(s) of the character variety containing characters of irreducible representations are birational to…
Researchers extend Godbillon-Vey functional to almost contact manifolds, finding critical structures.
problem Finding optimal almost contact manifolds using the Godbillon-Vey functional.
method Introduced a Godbillon-Vey type functional for 3D almost contact manifolds and found its Euler-Lagrange equations.
result Constructed critical 3D almost contact manifolds with double-twisted product structure.
This is the first of two companion papers in which a thorough study of the normal form and the first integrability conditions arising from {\em bi-conformal vector fields} is presented. These new symmetry transformations were introduced in {\em Class. Quantum Grav.} \textbf{21}, 2153-2177 and some of their basic proper…
New formalism solves kinematical constraints in curved backgrounds and non-trivial states.
problem Solving kinematical constraints due to Weyl invariance in curved backgrounds and non-trivial states.
method Constructing Weyl covariant geometric objects and identifying them as building blocks of correlation functions.
result Exact agreement with thermal OPEs and holographic computations for thermal 2-point functions.
Conjectures closed-form expressions and cyclotomic expansions for knot invariants.
problem Calculating HOMFLY-PT invariants of knots colored by rectangular diagrams.
method Interpolation Macdonald polynomials and cyclotomic expansions.
result Conjectured closed-form expressions and cyclotomic expansions for knot invariants.
The paper studies slopes for knot fillings with left-orderable fundamental groups.
problem Understanding slopes for knot fillings with left-orderable fundamental groups.
method Using the Riley polynomial and root analysis, the paper computes and conjectures on slopes.
result The paper computes the range of rational slope r for left-orderable fillings of two-bridge knots. We show that all twist knots, certain double twist knots and some other 2-bridge knots are minimal elements for the partial ordering on the set of prime knots. The key to these results are presentations of their character varieties using Chebyshev polynomials and a criterion for irreducibility of a polynomial of two va…
Let M be a two cusped hyperbolic 3-manifold and let M(r) be the result of r Dehn filling of a fixed cusp of M. We study canonical components of the SL(2,C) character varieties of M(r). We show that the gonality of these sets is bounded, independent of the filling parameter. We also obtain bounds, depending on r, for th…
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.
New method produces corks using Heegaard Floer homology.
problem Creating corks for certain knots.
method Floer-theoretic condition and formalism.
result Many new examples of corks produced.
The AJ conjecture relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. It has been verified for some classes of knots, including all torus knots, most double twist knots, (-2,3,6n \pm 1)-pretzel knots, and most cabled knots over torus knots. In this paper we study the AJ conjecture for (…
Our results concern geometry of a manifold endowed with a pair of complementary orthogonal distributions (plane fields) and a time-dependent Riemannian metric. The work begins with formulae concerning deformations of geometric quantities as the Riemannian metric varies conformally along one of the distributions. Then w…
Our goal is to compute the minimal-order recurrence of the colored Jones polynomial of the 7_4 knot, as well as for the first four double twist knots. As a corollary, we verify the AJ Conjecture for the simplest knot 7_4 with reducible non-abelian SL(2,C) character variety. To achieve our goal, we use symbolic summatio…
Detecting essential surfaces in 3-manifolds and orbifolds using character varieties.
problem Detecting essential surfaces in 3-manifolds and orbifolds.
method Extending Culler and Shalen's construction to 3-orbifolds using SL2(C) character variety. result Any slope detected on a canonical component of the (P)SL2(C) character variety of a one-cusped hyperbolic 3-manifold with symmetries must be the slope of a symmetric surface. Study on Jones polynomials and their roots in the unit circle and complex plane.
problem Understanding the roots of Jones polynomials for knots and links.
method Analyzing solutions of the equation JK(t)=1 for double-twist knots and links. result The set of solutions to JKn(t)=1 is dense in the unit circle and complex plane. We investigate the behavior of the SL(2,C) Casson invariant for 3-manifolds obtained by Dehn surgery along two-bridge knots. Using the results of Hatcher and Thurston, and also results of Ohtsuki, we outline how to compute the Culler--Shalen seminorms, and we illustrate this approach by providing explicit computations …
The study characterizes slopes for hyperbolic knots and Whitehead doubles.
problem Identifying characterizing slopes for hyperbolic knots and Whitehead doubles.
method Combining JSJ decompositions, geodesic lengths, and volume inequalities.
result Explicit conditions for characterizing slopes and examples of non-characterizing slopes.
The paper connects knot homology, quantum 6j-symbols, and complements of knots.
problem Investigating the relationship between knot homology, quantum 6j-symbols, and knot complements.
method Developed a grading rule for HOMFLY-PT and Kauffman homology, found relationships between A-polynomials, and conjectured closed-form expressions for quantum 6j-symbols and knot complements.
result Closed-form expressions for SO(N) quantum 6j-symbols and conjectured expressions for (a,t)-deformed F_K for knot complements.
We introduce the concept of bi-conformal transformation, as a generalization of conformal ones, by allowing two orthogonal parts of a manifold with metric $\G$ to be scaled by different conformal factors. In particular, we study their infinitesimal version, called bi-conformal vector fields. We show the differential co…
In this paper we continue the study of bi-conformal vector fields started in {\em Class. Quantum Grav.} {\bf 21} 2153-2177. These are vector fields defined on a pseudo-Riemannian manifold by the differential conditions $\lie P_{ab}=φP_{ab}$, $\lieΠ_{ab}=χΠ_{ab}$ where Pab, Πab are orthogonal and complementary…
Paper defines a new invariant for 4-manifolds with boundary.
problem Defining a new invariant for 4-manifolds with boundary.
method Defined the relative L-invariant by measuring path lengths in the cut complex of a trisection surface.
result Shows that if a rational homology ball has a zero relative L-invariant, it must be the 4-ball.
Survey para-Hermitian geometry and its applications in physics.
problem Capturing double field theory concepts on para-Hermitian manifolds.
method Geometric theory of Lagrangian and Hamiltonian systems, deformations of para-Kahler structures, non-linear connections, and weak integrability.
result Reproduce generalized fluxes in para-Hermitian geometry and describe their emergence.
The thesis examines the relationship between three-manifold invariants and knot theory, finding equalities and patterns.
problem Examining the relationship between three-manifold invariants and knot theory.
method Analytic continuation and quiver representation theory.
result Found equalities and patterns in knot theory and quiver representation.