New infinite families of twisted torus knots found.
problem Identifying new types of twisted torus knots.
method Finding new infinite families of twisted torus knots with a single negative twist.
result Eight new infinite families of twisted torus knots are discovered.
Formula for Alexander polynomial of twisted torus knots derived.
problem Calculating Alexander polynomial for a specific class of knots.
method Knot group presentation combined with Fox's calculus.
result Explicit formula for Alexander polynomial of twisted torus knots.
New families of twisted torus knots found with essential surfaces.
problem Whether all twisted torus knots have essential tori.
method Analyzing sequences of twists on torus knots.
result Found two new families of toroidal twisted torus knots.
New findings on T-links derived from torus links.
problem Difficulty in determining geometric type of T-links.
method Examined T-links obtained by full twists along torus links.
result T-links obtained by full twists are not torus links.
The paper classifies twisted torus links that are unlinks.
problem Identifying conditions for twisted torus links to be unlinks.
method Characterization of parameter families (p,q,r,s) for unlinking twisted torus links. result Complete characterization of parameter families for unlinking twisted torus links.
The study finds hyperbolic twisted torus links for certain twists.
problem Identifying hyperbolic twisted torus links.
method Analyzing (p,q)-torus links with r strands twisted s times. result All hyperbolic twisted torus links are identified for ∣s∣>3. Study bounds the Morse index of a special torus to 1.
problem Bounding the Morse index of a conformal harmonic torus.
method Min-max construction with harmonic replacement and conformal harmonic torus.
result The Morse index is bounded by one.
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).
Invariant for 3-manifolds with torus boundary defined.
problem Defining invariants for 3-manifolds with specific boundaries.
method Module over a weighted A-infinity algebra associated to a torus.
result Invariant constructed for bordered 3-manifolds with torus boundary.
For an arbitrary positive integer n and a pair (p,q) of coprime integers, consider n copies of a torus (p,q) knot placed parallel to each other on the surface of the corresponding auxiliary torus: we call this assembly a torus n-link. We compute economical presentations of knot groups for torus links using t…
Lee's work on twisted torus knots with Fibonacci parameters is extended to Horadam parameters.
problem Classifying twisted torus knots with Horadam parameters.
method Using recursive Horadam parameters to generalize Lee's work on Fibonacci parameters.
result Families of twisted torus knots with Horadam parameters are provided.
It is known that connected sums of positive torus knots are not concordant to L-space knots. Here we consider differences of torus knots. The main result states that the subgroup of the concordance group generated by two positive torus knots contains no nontrivial L-space knots other than the torus knots themselves…
Study concordance of alternating torus knots to L-space knots.
problem When are linear combinations of alternating torus knots concordant to L-space knots?
method Proved Allen's conjecture for alternating torus knots and established a necessary condition.
result Linear combinations of alternating torus knots are concordant to L-space knots if and only if they are a single torus knot.
Study calculates fundamental groups of torus knots using algebraic topology.
problem Calculating the fundamental group of torus knots.
method Algebraic topology and group theory.
result Computed fundamental groups of torus knots.
A torus manifold M is a 2n-dimensional orientable manifold with an effective action of an n-dimensional torus such that MT=∅. In this paper we discuss the classification of torus manifolds which admit an invariant metric of non-negative curvature. If M is a simply connected torus manifold which a…
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.
Graphs embeddable on torus and linklessly in 3D can be embedded linklessly in standard torus.
problem Embedding linklessly in a standard torus for graphs embeddable on torus and in 3D.
method Analyzing graphs of order 9 and below, showing linkless embedding in standard torus.
result For graphs of order 9 and below, linkless embedding in standard torus is possible.
We consider a surface link in the 4-space which can be presented by a simple branched covering over the standard torus, which we call a torus-covering link. Torus-covering links include spun T2-knots and turned spun T2-knots. In this paper we braid a torus-covering link over the standard 2-sphere. This gives an u…
A twisted torus knot is a knot obtained from a torus knot by twisting adjacent strands by full twists. The twisted torus knots lie in F, the genus 2 Heegaard surface for S3. Primitive/primitive and primitive/Seifert knots lie in F in a particular way. Dean gives sufficient conditions for the parameters of the tw…
Study shows surprising cobordism distances between certain torus knots.
problem Determining cobordism distances between thin and thick torus knots.
method Analyzes locally flat cobordisms between torus knots with small and large braid indices.
result Surprising fact about torus knots as cross-sections of almost minimal cobordisms.
Study finds infinite non-fibered twisted torus knots.
problem Identifying non-fibered twisted torus knots.
method Explicit formula for Alexander polynomial, leading coefficients analysis.
result Infinite families of non-fibered twisted torus knots found.
The study determines Z2-Thurston norms in Sol manifolds and embeds non-orientable surfaces.
problem Determining Z2-Thurston norms in Sol manifolds and embedding non-orientable surfaces. method Analyzing the action of torus maps on curve complexes and constructing incompressible surfaces.
result Determination of Z2-Thurston norms and embeddability of non-orientable surfaces in Sol manifolds. We introduce the notion of a local torus action modeled on the standard representation (for simplicity, we call it a local torus action). It is a generalization of a locally standard torus action and also an underlying structure of a locally toric Lagrangian fibration. For a local torus action, we define two invariants…
Study theta-curves on torus in 3-sphere, classifying them.
problem Classify theta-curves on torus in 3-sphere.
method Analyze nontrivial knots and essential arcs, compute constituent knots, identify structure.
result Complete classification of torus theta-curves up to isotopy and homeomorphism.
In this work, it is shown that a simply-connected, rationally-elliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to the number of odd-dimensional spherical factors in the product. As an a…
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.
We define a torus algebra for Heegaard Floer homology.
problem Developing algebraic structures for 3-manifold homology.
method Combinatorial and abstract algebraic constructions.
result Established connection to wrapped Fukaya category.
Classifies Legendrian and transverse torus knots in S3.
problem Classifying Legendrian and transverse torus knots in S3. method Complete coarse classification using Legendrian and transverse properties.
result Complete classification of torus knots in contact structures on S3. We give an affirmative answer to the Halperin-Carlsson conjecture for the homologically injective torus actions on closed manifolds. This class contains holomorphic torus actions on compact Kahler manifolds, torus actions on compact Riemannian flat manifolds.
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.
The paper proves conditions for 2-torus manifolds to be equivariantly formal.
problem Characterizing 2-torus manifolds as equivariantly formal.
method Proving 2-torus manifolds are equivariantly formal under specific conditions.
result 2-torus manifolds are equivariantly formal if and only if the action is locally standard and all faces of the orbit space are mod 2 acyclic.
Study fibrations over S2 with same singularities, showing monodromies are equivalent up to direct sums.
problem Classifying torus fibrations over S2 up to fibre sum stabilisation. method Analyzing monodromies and using direct sums with certain torus Lefschetz fibrations.
result Global monodromies of fibrations with same singularities are Hurwitz equivalent after direct sums.
We show that any non-minimal bridge decomposition of a torus knot is stabilized and that n-bridge decompositions of a torus knot are unique for any integer n. This implies that a knot in a bridge position is a torus knot if and only if there exists a torus containing the knot such that it intersects the bridge sphe…
Study signatures of torus links and their cores using Neumann's equivariant signatures and Hirzebruch's formula.
problem Computing signatures of torus links and their cores.
method Use Neumann's equivariant signatures and rewrite Hirzebruch's formula for torus links (without cores) in terms of integral points in a parallelogram.
result Rewritten Hirzebruch's formula for torus links with cores using integral points in a parallelogram.
In 2006 Masuda and Suh asked if two compact non-singular toric varieties having isomorphic cohomology rings are homeomorphic. In the first part of this paper we discuss this question for topological generalizations of toric varieties, so-called torus manifolds. For example we show that there are homotopy equivalent tor…
We introduce a new construction of a surface link in the 4-space. We construct a surface link as a branched covering over the standard torus, which we call a torus-covering link. We show that a certain torus-covering T2-link is equivalent to the split union of spun T2-links and turned spun T2-links. We show th…
Each closed oriented 3-manifold M is naturally associated with a set of integers D(M), the degrees of all self-maps on M. D(M) is determined for each torus bundle and torus semi-bundle M. The structure of torus semi-bundle is studied in detail. The paper is a part of a project to determine D(M) for all 3-ma…
Researchers compute gl2-skein modules for lens spaces.
problem Computing gl2-skein modules for lens spaces. method Action of gl2-skein algebra on solid torus's gl2-skein module. result Lens spaces' gl2-skein modules span by specific elements. Complete classification of rod complements in 3-torus using topology.
problem Classifying rod complements in the 3-torus.
method Topological arguments.
result Complete classification of all rod complements in the 3-torus.
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.
Characterizes Legendrian knots in lens spaces.
problem Classifying Legendrian knots in lens spaces.
method Splitting lens spaces and using convex Heegaard decomposition.
result All Legendrian torus knots in universally tight lens spaces are classified.
New invariant defined for unoriented knots, proving no factorization through topological concordance.
problem Defining and proving properties of unoriented slice-torus invariants.
method Introducing and proving properties of unoriented slice-torus invariants.
result Unoriented slice-torus invariants do not factor through the topological concordance group.
Jones polynomial for twisted torus knots is trivial if and only if the knot is trivial.
problem Computing Jones polynomial for a specific family of links.
method Computed Jones polynomial for T((p,q),(2,s)) links, focusing on s=2n. result Jones polynomial is trivial if and only if the knot is trivial.
In this note, we first classify all topological torus knots lying on the Heegaard torus in lens spaces, and then we study Legendrian representatives of these knots. We classify oriented positive Legendrian torus knots in the universally tight contact structures on the lens spaces up to contactomorphism.
We compute the triply graded Khovanov-Rozansky homology of a family of links, including positive torus links and Syml-colored torus knots.
Classifies symplectic torus actions up to equivariant symplectomorphism.
problem Classifying symplectic torus actions up to equivariant symplectomorphism.
method Classification theorems based on Duistermaat and Pelayo's work on symplectic torus actions with coisotropic orbits.
result Every almost isotropy-maximal symplectic torus action is equivariantly diffeomorphic to a product of a symplectic toric manifold and a torus.
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. We observe that the strong slope conjecture implies that the degree of the colored Jones polynomial detects all torus knots. As an application we obtain that an adequate knot that has the same colored Jones polynomial degrees as a torus knot must be a (2,q)-torus knot.