Study calculates virtually-cyclic dimension for specific mapping class groups.
problem Calculating virtually-cyclic dimension for mapping class groups of specific surfaces.
method Analyzes mapping class groups of punctured spheres with up to six punctures.
result Determines virtually-cyclic dimension for specific surfaces.
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.
Study shows differences of torus knots do not form L-space knots.
problem Understanding concordance of knots and their relation to L-space knots. method Examined differences of torus knots and their concordance properties.
result Subgroups generated by two positive torus knots contain no nontrivial L-space 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.
Computes knot groups for torus links using Seifert-van Kampen theorem.
problem Computing knot groups for torus links.
method Groupoid version of the Seifert-van Kampen theorem.
result Economical presentations of knot groups for 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.
Study homology of torus links and colored knots.
problem Computing homology of specific link types.
method Khovanov-Rozansky homology for positive torus links and colored knots.
result Computed homology for a family of links including torus links and colored knots.
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).
New examples of positive twisted torus knots with same surface slope found.
problem Finding new examples of positive twisted torus knots with same surface slope.
method Extending Dean's conditions and using new four-parameter family.
result Four-parameter family of positive twisted torus knots with same surface slope.
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.
The strong slope conjecture helps identify torus knots.
problem Detecting torus knots using colored Jones polynomials.
method Observation and application of the strong slope conjecture.
result An adequate knot with matching polynomial degrees is a (2,q)-torus knot. 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.
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…
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 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.
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. 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.
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.
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.
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…
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.
Sharp signature bound for 3-strand torus knots proved.
problem Determining the topological 4-genus of 3-strand torus knots.
method Using McCoy's twisting method and improving upper bounds.
result Signature bound is sharp for 3-strand knots and off by at most 1 for 4- and 6-strand knots.
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.
Classifies symplectic fillings of specific torus bundles.
problem Classifying strong and exact symplectic fillings of virtually overtwisted torus bundles.
method Using Menke's JSJ-type decomposition theorem and round symplectic 1-handle attachment.
result Conditions for distinct tight lens space fillings to yield the same torus bundle filling.
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…
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.
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. Study on sums of torus knots concordant to alternating knots.
problem Which sums of torus knots are concordant to alternating knots?
method Effective obstructions based on Heegaard Floer homology.
result Described some effective obstructions for sums of two torus knots.
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…
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.
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.