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.
Extends field theory foundations to infinitesimal spaces, simplifying complex concepts.
problem Develop rigorous foundations for field theory, especially for infinitesimal spaces.
method Formulates local Lagrangian field theory in a new category of thickened smooth sets.
result Establishes a firm foundation for field theory, including tangent bundles and perturbative considerations.
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…
Analog of Kauffman bracket for non-orientable knots in thickened surface.
problem Defining an invariant for non-orientable knots in a non-orientable surface.
method Proposes an analog of the Kauffman bracket polynomial with modified sign rules.
result Polynomial is an isotopy invariant and independent of classical Kauffman for orientable covers.
We construct smooth 4-manifolds that are homeomorphic but not diffeomorphic to the "cusp" and the "fishtail", which are certain thickened singular 2-spheres.
We define additional gradings on two generalisations of Khovanov homology (one due to the first author, the other due to the second), and use them to define invariants of various kinds of embeddings. These include invariants of links in thickened surfaces and of surfaces embedded in thickened 3-manifolds. In particul…
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…
We study the atomic embeddability testing problem, which is a common generalization of clustered planarity (c-planarity, for short) and thickenability testing, and present a polynomial-time algorithm for this problem, thereby giving the first polynomial-time algorithm for c-planarity. C-planarity was introduced in 1995…
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.
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.
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.
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.
Given a sample Y from an unknown manifold X embedded in Euclidean space, it is possible to recover the homology groups of X by building a Vietoris--Rips or Čech simplicial complex on top of the vertex set Y. However, these simplicial complexes need not inherit the metric structure of the manifold, in particular…
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.
Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the geometry of circle actions on Euclidean space, the structure of zeros of trigonometr…
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.
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.
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.
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.
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.
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.
Almost forty years ago, C.T.C. Wall systematically analyzed the set of "thickenings" of a finite CW complex. Of the results he obtained, probably the most computationally important is the "suspension theorem," which is an exact sequence relating the n-dimensional thickenings of a finite complex to its (n+1)-dimensional…
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.
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.
The HKR (Hennings-Kauffman-Radford) framework is used to construct invariants of 4-thickenings of 2-dimensional CW complexes under 2-deformations (1- and 2- handle slides and creations and cancellations of 1-2 handle pairs). The input of the invariant is a finite dimensional unimodular ribbon Hopf algebra A and an elem…
Establishes bounds on Andrews-Curtis moves for trivial group presentations.
problem Understanding presentations of the trivial group and Andrews-Curtis moves.
method Explicit upper bounds on stable Andrews-Curtis moves for thickenable presentations.
result Thickenable presentations of the trivial group satisfy the Andrews-Curtis conjecture.
A group invariant for links in thickened closed orientable surfaces is studied. Associated polynomial invariants are defined. The group detects nontriviality of a virtual link and determines its virtual genus.
This paper classifies periodic weaves and their universal cover, extending Tait's conjectures.
problem Classifying periodic weaves and their universal cover in thickened surfaces.
method Introducing hyperbolic periodic weaves, extending Tait's conjectures, and using a generalized Kauffman bracket polynomial.
result Tait's conjectures are extended to minimal reduced alternating weaving motifs.
New algebra invariant distinguishes Legendrian knots in convex surfaces.
problem Distinguishing Legendrian knots in convex surfaces using invariants.
method Defined a differential graded algebra (DGA) for Legendrian knots in thickened convex surfaces, generating it from Reeb chords and counting immersed polygons.
result The stable tame isomorphism type of the DGA is invariant under Legendrian isotopy and can distinguish knots not distinguishable by classical invariants.
We introduce mappings between spaces of functions on (super)manifolds that generalize pullbacks with respect to smooth maps but are, in general, nonlinear (actually, formal). The construction is based on canonical relations and generating functions. (The underlying structure is a formal category, which is a "thickening…
Extends Heisenberg homology to ribbon graphs.
problem Configurations in bounded surfaces.
method Regular thickening of ribbon graphs.
result Heisenberg homology applied to ribbon graphs.
The paper develops a theory of skein adequate links in thickened surfaces and proves Tait conjectures.
problem Establishing Tait conjectures for adequate links in thickened surfaces.
method Applying Kauffman bracket skein algebras to develop a theory of skein adequate links and proving Tait conjectures.
result The crossing number is additive under connected sum for adequate links in thickened surfaces.
We construct a compact, contractible 4-manifold C, an infinite-order self-diffeomorphism f of its boundary, and a smooth embedding of C into a closed, simply connected 4-manifold X, such that the manifolds obtained by cutting C out of X and regluing it by powers of f are all pairwise nondiffeomorphic. The…
We introduce two tools, dynamical thickening and flow selectors, to overcome the infamous discontinuity of the gradient flow endpoint map near non-degenerate critical points. More precisely, we interpret the stable fibrations of certain Conley pairs (N,L), established in [2,3], as a dynamical thickening of the stable…
A cobordism between links in thickened surfaces consists of a surface S and a 3-manifold M, with S properly embedded in M×I. We show that there exist links in thickened surfaces such that if (S,M) is a cobordism between them in which S is simple, then M must be complex. That is, there…
We prove a Kauffman-Murasugi-Thistlethwaite theorem for alternating links in thickened surfaces. It states that any reduced alternating diagram of a link in a thickened surface has minimal crossing number, and any two reduced alternating diagrams of the same link have the same writhe. This result is proved more general…
New findings on hyperbolicity of augmented links in thickened surfaces.
problem Proving hyperbolicity of links in thickened surfaces.
method Extending hyperbolicity results to generalized augmented cellular alternating links.
result Generalized augmented cellular alternating links in thickened surfaces are hyperbolic.
An algebraic system is proposed that represent surface cobordisms in thickened surfaces. Module and comodule structures over Frobenius algebras are used for representing essential curves. The proposed structure gives a unified algebraic view of states of categorified Jones polynomials in thickened surfaces and virtual …
This paper defines a functor for L∞-modules and applies it to Khovanov homology.
problem Defining a functor for L∞-modules and proving its properties. method Abstract approach using morphisms of L∞-algebras. result Restriction of scalars defines a functor between L∞-modules. Decomposes string links in a surface into prime components.
problem Decomposing string links in a surface into prime components.
method Proves a prime decomposition theorem for string links in a thickened surface.
result Any non-braid string link can be uniquely decomposed into prime string links up to braid equivalence.
Study links' flat-virtual diagrams to create link invariants.
problem Equivalence and invariants of flat-virtual diagrams.
method Maps from links in thickened surfaces to flat-virtual links.
result Investigation of flat-virtual diagrams' equivalence and invariants.
Study invariants of null-homologous knots in thickened surfaces.
problem Understanding concordance properties of knot invariants.
method Using Gordon-Litherland pairing and cobordism results for closed unoriented surfaces.
result Brown invariants are invariant under concordance of spanning surfaces.
Algorithm converts plat to standard closure of braids in 3D and related spaces.
problem Converting plat to standard closure of braids in different spaces.
method Algorithmic approach for plat to standard closure conversion in \(\mathbb{R}^3\), handlebodies, and thickened surfaces.
result Algorithm is quadratic for plat to standard closure and linear for standard to plat closure.
The study determines conditions for hyperbolicity of links in thickened surfaces with boundary.
problem Conditions for hyperbolicity of links in thickened surfaces with boundary.
method Simple conditions and embeddings of surfaces in ambient manifolds.
result Conditions guaranteeing hyperbolicity of links in thickened surfaces and fiber bundles.
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.
Given a sample of points X in a metric space M and a scale r>0, the Vietoris-Rips simplicial complex VR(X;r) is a standard construction to attempt to recover M from X up to homotopy type. A deficiency of this approach is that VR(X;r) is not metrizable if it is not locally finite, and thu…
New invariants for virtual knots via spanning surfaces.
problem Tackling invariants for virtual knots and distinguishing them from classical knots.
method Defining new invariants using spanning surfaces and Seifert matrices.
result An Alexander polynomial for virtual knots that can distinguish knots from their mirrors.