Hyperbolic links in thickened torus decompose into angled tetrahedra.
problem Hyperbolicity of links in thickened torus.
method Decomposition into torihedra, angled pyramids, and angled tetrahedra.
result Augmented links in thickened torus are hyperbolic.
Study fully augmented links in thickened torus, generalizing S3 results.
problem Classify and describe geometric properties of fully augmented links in thickened torus.
method Geometric analysis and decomposition of link complements into ideal right-angled torihedra.
result Proves Volume Density Conjecture for fully augmented links in thickened torus.
The paper extends knotoid theory to annular and toroidal settings.
problem Extending knotoid theory to new geometric settings.
method Introducing new knotoid types, extending bracket polynomials.
result Universal bracket polynomials for annular and toroidal knotoids.
Defines homotopy type for links in thickened surfaces.
problem Homotopical Khovanov homology of links in higher genus surfaces.
method Stable homotopy type for links in thickened torus and higher genus surfaces.
result Definition of Khovanov-Lipshark-Sarkar homotopy type for links in thickened surfaces.
The paper develops a method to map knots in a cylinder to virtual-flat knots.
problem How to map knots in a cylinder to virtual knots.
method Construct a diagram on a cylinder with invisible crossings, then pull back invariants.
result Developed a method to map knots in a cylinder to virtual-flat knots.
A new polynomial invariant for links in a thickened torus exhibits volume conjecture behavior.
problem Defining and characterizing a new polynomial invariant for links in a thickened torus.
method Defining a new invariant JnT, proving properties, and providing constructions. result The invariant JnT exhibits volume conjecture behavior, providing the first example of this in a virtual link. New TQFTs distinguish torus bundles and lens spaces.
problem Existence of faithful (2+1)-TQFTs.
method Subcategory of 3-cobordisms containing disjoint unions of tori and simplest cobordisms between them, defined TQFTs.
result TQFTs can distinguish torus bundles and lens spaces.
The paper extends knot polynomials to annular and toroidal pseudo links.
problem Analyzing pseudo links with undefined crossings.
method Introducing Kauffman bracket and Jones-type polynomials for annular and toroidal pseudo links.
result New tools for studying annular and toroidal pseudo links.
New homotopy types defined for links in thickened surfaces with higher genus.
problem Defining stable homotopy types for links in surfaces with higher genus.
method Defined Khovanov-Lipshitz-Sarkar homotopy types and Steenrod squares for links in thickened surfaces with genus > 1.
result First meaningful Khovanov-Lipshitz-Sarkar stable homotopy types for links in 3-manifolds other than the 3-sphere.
We prove a Torres-like formula for the L2-Alexander torsions of links, as well as formulas for connected sums and cablings of links. Along the way we compute explicitly the L2-Alexander torsions of torus links inside the three-sphere, the solid torus and the thickened torus.
A knot K is called Gordian adjacent to a knot L if there exists an unknotting sequence for L containing K. We provide a sufficient condition for Gordian adjacency of torus knots via the study of knots in the thickened torus. We also completely describe Gordian adjacency for torus knots of index 2 and 3 using Levine-Tri…
Study spaces of knots and links in specific 3-manifolds.
problem Determine the homotopy types of spaces of knots and links in various 3-manifolds.
method Recursive determination of homotopy types, using fundamental groups and quotient spaces.
result Homotopy types of spaces of knots in solid torus and thickened torus are determined.
Totally geodesic surfaces found in knots and links.
problem Finding totally geodesic surfaces in knots and links.
method Constructing infinite families of knots and links with totally geodesic spanning surfaces in various 3-manifolds.
result Infinite families of knots and links with totally geodesic spanning surfaces in multiple 3-manifolds.
The paper extends knot theory to annular and toroidal pseudo knots.
problem Defining and classifying pseudo knots in annular and toroidal settings.
method Introducing pseudo knots as equivalence classes under moves, lifting to torus, and exploring inclusion relations.
result New invariants for classifying pseudo knots and links in solid and thickened torus.
Proves NP and co-NP status for knot core recognition in solid torus.
problem Determining if a knot is the core of a solid torus.
method Alternate proof and corollary of Hopf link recognition problem.
result Proves NP and co-NP status for solid torus core recognition problem.
To a closed braid in a solid torus we associate a trace graph in a thickened torus in such a way that closed braids are isotopic if and only if their trace graphs can be related by trihedral and tetraherdal moves. For closed braids with a fixed number of strands, we recognize trace graphs up to isotopy and trihedral mo…
Study links in knitted textiles using knot theory.
problem Identify and classify links in knitted textiles.
method Correspondence between links in thickened torus and 2-periodic weft-knitted textiles, using link invariants and ribbon knots.
result New stitch patterns and links in knitted textiles identified.
To an oriented link in a solid torus we associate a trace graph in a thickened torus in such a way that links are isotopic if and only if their trace graphs can be related by moves of finitely many standard types. The key ingredient is a study of codimension~2 singularities of link diagrams. For closed braids with a fi…
Paper introduces flat-virtual knots and invariants for classical knots.
problem Constructing a map from classical knots to virtual knots.
method Definition of flat-virtual knots and invariants (Alexander-like polynomial, Kauffman bracket).
result Introduction of flat-virtual knots and their invariants.
In this paper we classify Legendrian and transverse knots in the knot types obtained from positive torus knots by cabling. This classification allows us to demonstrate several new phenomena. Specifically, we show there are knot types that have non-destabilizable Legendrian representatives whose Thurston-Bennequin invar…
The paper explores group presentations for links in thickened surfaces, proving their relationship and introducing new invariants.
problem Proving the relationship between group presentations for links in thickened surfaces.
method Combining combinatorial arguments and homological information from surfaces to establish the relationship and introduce new invariants.
result The relationship between Dehn presentations and abelian Dehn coloring groups, and the introduction of the module C as a stronger invariant. We provide examples of towers of covers of cusped hyperbolic 3-manifolds whose exponential homological torsion growth is explicitly computed in terms of volume growth. These examples arise from abelian covers of alternating links in the thickened torus. A corollary is that the spanning tree entropy for each regular pla…
A combinatorial framework classifies genus-one knots and links.
problem Classifying knots and links on a torus.
method Maps on surfaces, permutation pairs, bit data, Reidemeister II reductions, state sums.
result Completely enumerated and reproducible genus-one knot and link diagrams.
Based on the presentation of the Kauffman bracket skein module of the torus given by the third author in previous work, Charles D. Frohman and Răzvan Gelca established a complete description of the multiplicative operation leading to a famous product-to-sum formula. In this paper, we study the multiplicative structure …
We define the notion of a knot type having Legendrian large cables and show that having this property implies that the knot type is not uniformly thick. Moreover, there are solid tori in this knot type that do not thicken to a solid torus with integer sloped boundary torus, and that exhibit new phenomena; specifically,…
Knitted and woven textile structures are examples of doubly periodic structures in a thickened plane made out of intertwining strands of yarn. Factoring out the group of translation symmetries of such a structure gives rise to a link diagram in a thickened torus. Such a diagram on a standard torus is converted into a c…
This paper studies the homotopy type of the moduli space of compact n-manifold thickenings of a finite complex. The main result computes the homotopy fibers of the stabilization map from n-thickenings to (n+1)-thickenings in a wide range. In particular, we obtain an EHP-type sequence for thickenings extending work of C…
This paper studies isotopies of periodic tangles in 3-manifolds using finite covers.
problem Understanding equivalence relations of periodic tangles in 3-manifolds.
method Employing techniques from 3-manifold topology to study complements of links.
result Isotopies between different finite covers of isotopic links are preserved.
The main purpose of this article is to classify contact structures on some 3-manifolds, namely lens spaces, most torus bundles over a circle, the solid torus, and the thickened torus T^2 x [0,1]. This classification completes earlier work (by Etnyre [math.DG/9812065], Eliashberg, Kanda, Makar-Limanov, and the author) a…
Paper constructs motifs from planar tilings for DP weaves and polycatenanes.
problem Creating complex entangled structures from periodic tilings.
method Combinatorial methodology using polygonal link transformations.
result Predicting the type of motif from a given tiling and polygonal link method.
Geometrically interprets virtual knotoids in thickened surfaces.
problem No specific problem stated; focuses on geometric interpretation.
method Geometric interpretation of virtual knotoids as arcs in thickened surfaces.
result Shows virtual knotoid theory as a generalization of classical knotoid theory.
Homotopy types of Vietoris-Rips metric thickenings of the circle confirmed.
problem Understanding the homotopy types of Vietoris-Rips metric thickenings of the circle.
method Finding quotients of the metric thickenings that preserve homotopy type and showing that the quotient spaces can be described as CW complexes.
result The Vietoris-Rips metric thickenings of the circle are homotopy equivalent to odd-dimensional spheres at the expected scale parameters.
Paper defines and analyzes mathematical equivalence of periodic tangles.
problem Understanding the equivalence of periodic tangles.
method Established mathematical framework, characterized isotopies, generalized results.
result Characterization of DP tangle equivalence based on motifs.
Characterizes alternating links in thickened surfaces using Gordon-Litherland pairing.
problem Identifying alternating links in thickened surfaces.
method Extension of Gordon-Litherland pairing to thickened surfaces.
result A non-split link in a thickened surface is alternating if and only if it bounds two definite surfaces of opposite sign.
We consider the representation space of a compact surface, that is the space of morphisms from the fundamental group to SU(2) up to conjugation. We show that the trace functions associated to multicurves on the surface are linearly independent as functions on the representation space. The proof relies on the Fourier de…
Paper characterizes isotopies and hyperbolicity of weaves using geodesics.
problem Characterizing isotopies and hyperbolicity of weaves.
method Using diagrams of closed geodesics and normal positions of essential surfaces.
result Weaves are hyperbolic and cannot have essential Conway spheres.
This paper classifies a specific weave type by their crossing number.
problem Classifying doubly periodic untwisted (p,q)-weaves.
method Classification by crossing number, introducing crossing matrix for equivalence.
result Classification of untwisted (p,q)-weaves by their crossing number.
The paper studies alternating links in thickened surfaces using flow lattices and disc mutations.
problem Understanding alternating links in thickened surfaces and their invariants.
method Using integer flows on Tait graphs and disc mutations, the paper proves invariants and compares link properties.
result Found alternating knots with isometric flow lattices but different linking forms.
Meridian lemma extended to fully alternating links in thickened surfaces.
problem Extending Menasco's meridian lemma to fully alternating links in thickened surfaces.
method Developed a new meridian lemma for fully alternating links in thickened orientable surfaces of positive genus.
result The meridian lemma holds for fully alternating links in thickened surfaces.
New Khovanov homology for links with multiple punctures.
problem Defining a new Khovanov homology for links with multiple punctures.
method Defined a variant of Khovanov homology for links in thickened disks with multiple punctures, related to previous work by spectral sequences.
result Spectral sequences recover annular Khovanov homology to Khovanov homology.
A knot in a thickened surface K is a smooth embedding K:S1→Σ×[0,1], where Σ is a closed, connected, orientable surface. There is a bijective correspondence between knots in S2×[0,1] and knots in S3, so one can view the study of knots in thickened surfaces as an extension of classic…
This paper generalizes octahedral decomposition to links in thickened surfaces.
problem Understanding the geometry of links in thickened surfaces.
method Octahedral decomposition of links in thickened surfaces.
result Nonpositive curvature of the complement and essential-ness of edges proved.
Homotopy equivalence shown between complex and thickened versions of manifolds.
problem Homotopy equivalence between manifold complexes and thickened versions.
method Natural bijections and homotopy equivalences of Vietoris-Rips and Čech complexes and thickened versions.
result Natural bijections between complexes and thickened versions are homotopy equivalences.
The elliptic associator of Enriquez can be used to define an invariant of tangles embedded in the thickened torus, which extends the Kontsevich integral. This construction by Humbert uses the formulation of categories with elliptic structures. In this work we show that an extension of the LMO functor also leads to an e…
Spaces over BO are equivalent to thickened manifolds.
problem Understanding embeddings of manifolds in higher dimensions.
method Formal identification of manifolds with their thickened versions, using geometric constructions.
result The infinity-category of thickened smooth manifolds is equivalent to the infinity-category of finite spaces over BO.
A chord index homomorphism for knots in thickened surfaces is constructed.
problem Knot invariants in thickened surfaces.
method Constructing a chord index homomorphism from a subgroup of H1(Σ,Z) to chord indices of a knot K in ΣimesI. result Derived knot invariants from the homomorphism.
Extends tangle theory to include undetermined crossings in periodic structures.
problem Classical tangle theory's limitations in handling undetermined crossings.
method Introduces pseudo DP tangles, defined as liftings of pseudo motifs in the thickened torus, and analyzes them through diagrammatic methods.
result Defines equivalence for pseudo DP tangles and proves an analogue of Reidemeister theorem.
We compose the table of knots in the thickened torus T x I having diagrams with at most 4 crossings. The knots are constructed by the three-step process. First we list regular graphs of degree 4 with at most 4 vertices, then for each graph we enumerate all corresponding knot projections, and after that we construct the…