Simplified expressions for torus relations found.
problem Finding simplified expressions for torus relations.
method Explicit factorization of boundary multi-twists as products of Dehn twists.
result Simplified expressions for the k-holed torus relations. Abstract: Holographic principle applied to knots and links.
problem Understanding polynomial invariants and skein relations for knots and links.
method Formulated holographic principle for knots and links; used recurrence relations for polynomial invariants of torus knots and links.
result Derivation of Jones skein relation and its generalization.
Study the geometry of torus link character varieties, finding unexpected relations.
problem Understanding the geometry of torus link character varieties.
method Developed an intrinsic stratification to relate the geometry with torus knots, computed E-polynomial for SL2(C) and SL3(C).
result Unexpected relation with the number of strands of the link.
Researchers confirm a relation between knot invariants and provide formulas for torus knots.
problem Confirming a relation between knot invariants and providing formulas.
method Explicit formulas and algorithms for certain ADO-invariants of torus knots obtained from the series invariant of knot complements.
result Explicit formulas and algorithms for certain ADO-invariants of torus knots.
New recursive relation found for a specific torus knot.
problem Finding a recursive relation for a specific torus knot.
method Extending colored Jones polynomials to knots in (2p+1,2) torus knot complements and examining a particular knot. result An analogous recursive relation exists for a specific (2p+1,2) torus knot. The AJ conjecture, formulated by Garoufalidis, relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. It has been confirmed for all torus knots, some classes of two-bridge knots and pretzel knots, and most cabled knots over torus knots. The strong AJ conjecture, formulated by Sikora, relat…
New R-equivalence classes found for torus knot diagrams.
problem Classifying colorings of torus knots.
method Introducing R-equivalence relation on quandle colorings.
result Determined R-equivalence classes for RotE2-colorings of torus knots. New invariant defined for tied links in solid torus.
problem Defining an invariant for tied links in solid torus.
method Using skein relations and Jones' method over bt-algebra of type B with Markov trace.
result Recovery of invariant defined for tied links in solid torus.
New method computes triply graded link homology for torus links.
problem Computing triply graded link homology for torus links.
method Introducing a new method for triply graded link homology specifically adapted to torus links.
result Exact answers for (n,n)-torus links, verifying conjectures about homology and Hilbert schemes.
Study torus orbifolds with two fixed points and their cohomology.
problem Understanding the topological and cohomological properties of torus orbifolds with two fixed points.
method Analyze the equivariant topological type and use results from [DKS] to compute integral equivariant cohomology.
result Results on generators and relations for the cohomology of torus orbifolds with two fixed points.
We give definitions of cohomology determinants for compact, connected, orientable 3-manifolds. We also give formulae relating cohomology determinants before and after gluing a solid torus along a torus boundary component. Cohomology determinants are related to Turaev torsion, though the author hopes that they have othe…
Characterizes values of slice-torus invariants related to knot genus.
problem Understanding the values of slice-torus invariants for knots.
method Characterization based on stable smooth slice genus.
result Existence of slice torus invariants without explicit constructions.
L. Paoluzzi constructed a family of compact orientable three-dimensional hyperbolic manifolds with totally geodesic boundary, which were, by construction, closely related to the three-dimensional torus. This paper gives their complete classification up to isometry, and also their isometry groups. The key tool is the so…
The paper calculates intertwiners for a torus and proves a conjecture about their limits.
problem Relating quantum invariants to hyperbolic geometry using intertwiners.
method Explicit calculation of intertwiners for a closed torus and periodic diffeomorphisms.
result The limit superior of the trace of intertwiners is zero for certain diffeomorphisms.
Determine pairs of torus knots with genus one cobordisms, with exceptions.
problem Identify pairs of torus knots with genus one cobordisms.
method Combine obstructions from Heegaard Floer knot complex with explicit constructions.
result Determine pairs of torus knots with genus one cobordisms, with exceptions.
Let G be a simple complex algebraic group and g its Lie algebra. We show that the g-Witten-Reshetikhin-Turaev quantum invariants determine a deformation-quantization, C_q[X_G(torus)], of the coordinate ring of the G-character variety of the torus. We prove that this deformation is in the direction of the Goldman's brac…
Paper proves mirror symmetry for conifold transitions of torus knots.
problem Relating open Gromov-Witten invariants to topological recursion.
method Uses topological recursion on spectral curves.
result Proves mirror symmetry conjecture for conifold transitions.
Twisted torus knots and links are given by twisting adjacent strands of a torus link. They are geometrically simple and contain many examples of the smallest volume hyperbolic knots. Many are also Lorenz links. We study the geometry of twisted torus links and related generalizations. We determine upper bounds on their …
We compute the Heegaard Floer homology of S13(K) (the (+1) surgery on the torus knot Tp,q) in terms of the semigroup generated by p and q, and we find a compact formula (involving Dedekind sums) for the corresponding Ozsvath--Szabo d-invariant. We relate the result to known knot invariants of Tp,q as …
Study nearly parallel G2-structures with torus symmetry using multi-moment maps.
problem Characterize and construct nearly parallel G2-structures with torus symmetry.
method Use multi-moment map techniques and analyze the geometry of the base spaces.
result Locally, the construction may produce examples with four-torus symmetry.
This paper provides the technical details of gradient flow construction and related problems, which are essential for our construction of Lagrangian torus fibrations for Calabi-Yau hypersurfaces.
Study earthquake deformations on a once-punctured torus.
problem Understanding earthquake deformations on Teichmüller space.
method Two methods: linear recurrence relations and hyperbolic geometry.
result Algebraic and geometric interpretations of earthquake deformations.
Study of panhandle polynomials of torus links with geometric applications.
problem Characterizing the HOMFLY-PT polynomial of torus knots and links.
method Utilizing quantum group representations and the Rosso-Jones formula.
result Established panhandle-like structure of HOMFLY-PT polynomials for torus knots and links.
Formula connects linking number to spectral theory on 3-torus.
problem Computing linking number of multi-geodesics on 3-torus.
method Spectral theory of Laplace operator on differential forms.
result Formula for linking number of multi-geodesics on 3-torus.
New theorem for 4D links simplifies characterisation problem.
problem Long-standing open problem in link characterisation.
method Reidemeister Theorem for solid ribbon torus links.
result Complete characterisation of a related class of links.
Abstract: Topological quantum field theory connects graph evaluations to polynomial identities.
problem Graph evaluations in topological quantum field theory.
method Relates SO(3) topological quantum field theory trace evaluations to topological Tutte polynomial evaluations.
result Generalizes the Tutte golden identity for graphs on the torus.
Research extends geodesic length function study to three holed sphere.
problem Geodesic length function on orbifolds.
method Extending previous work on punctured torus to three holed sphere and related orbifolds.
result Extension to three holed sphere and related orbifolds.
We conjecture that the stable Khovanov homology of torus knots can be described as the Koszul homology of an explicit non-regular sequence of quadratic polynomials. The corresponding Poincare series turns out to be related to the Rogers-Ramanujan identity.
Study of quaternion Azumaya algebras in hyperbolic once-punctured torus bundles.
problem Character variety and topological invariants of Dehn fillings.
method Extension problem for quaternion Azumaya algebras.
result New phenomena in extension problem not seen in knot complements.
We define composite DAHA-superpolynomials of torus knots, depending on pairs of Young diagrams and generalizing the composite HOMFLY-PT polynomials in the theory of the skein of the annulus. We provide various examples. Our superpolynomials extend the DAHA-Jones (refined) polynomials and satisfy all standard symmetries…
New method calculates number of components in twisted torus links.
problem Determining the number of components of twisted torus links.
method Euclidean algorithm-like procedure based on parameters.
result Number of components is a multiple of gcd(p, q, r, s).
Characterizes regular parallelisms in 3D space with 2-torus action.
problem Characterizing regular parallelisms in 3D space with 2-torus action.
method Characterization using compactness, equivalence relations, and properties of complex vector spaces.
result There is a 1-dimensional subtorus fixing every parallel class, leading to 2- or 3-dimensional regular parallelisms.
Study shows convergence of certain metrics to flat torus.
problem Stability of metrics on three-torus with negative scalar curvature.
method Defined metrics and used Stern's inequality to show convergence.
result Subsequence of metrics converges to flat metric.
We prove that functions defined on a lattice in a finite dimensional torus with bounded finite differences can be smoothly extended to the whole torus, and relate the bounds on the extension's derivatives with bounds on the original function's finite differences.
We obtain an asymptotic formula for the eigenvalue distribution function of the Laplace-Beltrami operator on the two-dimensional torus in the adiabatic limit given by a Kronecker foliation. Related problems in number theory are discussed.
We derive formulas for HOMFLY polynomials of torus links using braid groups and linear recurrences.
problem Calculating HOMFLY polynomials for torus links.
method Using braid groups and linear recurrences, derived from the skein relation.
result Explicit formulas for HOMFLY polynomials of torus links T(3,n) and T(−3,n) are derived. A symmetric tensor field on a Riemannian manifold is called Killing field if the symmetric part of its covariant derivative is equal to zero. There is a one to one correspondence between Killing tensor fields and first integrals of the geodesic flow which depend polynomially on the velocity. Therefore Killing tensor fi…
New Garside structures found for torus knot groups and related braid groups.
problem Finding Garside structures for torus knot groups and related braid groups.
method Introducing a new Garside monoid M(n,m) for (n,m)-torus knot groups and other braid groups. result New Garside structures for (n,m)-torus knot groups and related braid groups are constructed. The noncommutative A-ideal of a knot is a generalization of the A-polynomial, defined using Kauffman bracket skein modules. In this paper we show that any knot that has the same noncommutative A-ideal as the (2,2p+1)-torus knot has the same colored Jones polynomials. This is a consequence of the orthogonality relation,…
We compute the vacuum expectation values of torus knot operators in Chern-Simons theory, and we obtain explicit formulae for all classical gauge groups and for arbitrary representations. We reproduce a known formula for the HOMFLY invariants of torus links and we obtain an analogous formula for Kauffman invariants. We …
Study Morse models for torus algebra related to knot homology.
problem Understanding algebraic structures of tori and knots.
method Construct Morse models and use multiple time scale dynamics.
result Identifies Cord(T_K) with Cord(K) and relates to Legendrian contact homology.
Study Ricci flow on torus bundles and related manifolds.
problem Compute Ricci flow formulas for torus bundles.
method Use invariant metrics compatible with connections on principal G-bundles.
result Solutions to Ricci flows on Heisenberg groups and Berger 3-spheres.
Proves conjecture about integer sums of torus knot torsions.
problem Integrality of sums of (g-1)st powers of adjoint Reidemeister torsions for torus knots.
method Introduced Verlinde numbers from modular S-matrix, proved integrality through recursion formulas.
result Proven integrality of sums of (g-1)st powers of adjoint Reidemeister torsions for all torus knots and non-negative g.
Formula calculates invariant for 3-manifolds with torus boundaries.
problem Calculating an invariant for 3-manifolds with torus boundaries.
method Generalized Chern-Simons invariant and provided a gluing formula.
result A gluing formula for the invariant d(M,ρ). Paper develops knot invariants for long knots in a torus.
problem Understanding long knots in a torus.
method Uses picture-valued and free group valued invariants.
result Developed powerful and easy to compare knot invariants.
We study the asymptotics of the higher dimensional Reidemeister torsion for torus knot exteriors, which is related to the results by W. Müller and P. Menal-Ferrer and J. Porti on the asymptotics of the Reidemeister torsion and the hyperbolic volumes for hyperbolic 3-manifolds. We show that the sequence of log |the high…
New knot homology invariant grows exponentially with color.
problem Constructing and understanding colored torus knot homology.
method Invariant construction and recursive formula for reduced HOMFLY homology.
result Doubly-graded invariant of positive torus knots grows exponentially in color.
The main result we give in this brief note relates, under suitable hypotheses, the φ-null Osserman, the null Osserman and the classical Osserman conditions to each other, via semi-Riemannian submersions as projection maps of principal torus bundles arising from a Lorentzian S-manifold.