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.
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.
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.
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.
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.
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.
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.
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.
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.
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 …
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…
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…
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 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.
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.
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…
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.
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.
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.
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 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.
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.
We construct an infinite family of homology theories of framed links in thickened surfaces, as well as a homology theory whose graded Euler characteristic is exactly the Kauffman bracket of the link in the surface. Both theories are based on ideas coming from Asaeda, Przytycki and Sikora's categorification of the Kauff…
New tool lassos connects virtual and surface link diagrams, changing primeness rules.
problem Establishing a new correspondence between virtual and surface link diagrams.
method Using a new tool called lassos, establishing a new correspondence between cellular link diagrams on closed surfaces and equivalence classes of virtual link diagrams.
result Traditional primeness for virtual links is stricter than for links in thickened surfaces.
Extends Gordon-Litherland pairing to links in thickened surfaces, defining new invariants.
problem Defining invariants for links in thickened surfaces.
method Extending Gordon-Litherland pairing, defining new invariants based on spanning surfaces.
result Invariants depend only on S∗-equivalence class of spanning surfaces and give well-defined invariants of virtual links. We introduce a relation of cobordism for knots in thickened surfaces and study cobordism invariants of such knots.
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.
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. Detects knots in thickened surfaces using instanton homology.
problem Detecting knots in thickened surfaces using homology.
method Uses Asaeda-Przytycki-Sikora (APS) homology and sutured instanton homology.
result Detects the unknot in (−1,1)imesΣ and characterizes minimal sutured instanton homology. 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.
This paper studies knots in a thickened surface and introduces a new way to label crossings.
problem Analyzing knots in a thickened surface with a new labeling system.
method Introducing diagrams, moves, and a new labeling system for knots in SgimesS1. result Developed a new method to label crossings in knots in SgimesS1. Algorithm calculates Jones polynomial from Goeritz matrix.
problem Calculating Jones polynomial from link diagrams.
method Explicit algorithm using Goeritz matrices.
result Jones polynomial can be recovered from orientable checkerboard surfaces.
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.
The flyping theorem is extended to virtual links and surfaces.
problem Proving the flyping theorem for virtual links and surfaces.
method Adapting geometric proof for thickened surfaces and using diagrammatic correspondence.
result Established an isomorphism between Gordon-Litherland pairing and 4-manifold intersection form.
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…
We modify the definition of the Khovanov complex for oriented links in a thickening of an oriented surface to obtain a triply graded homological link invariant with a new homotopical grading.
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.
Study links in 3-manifolds, linking volume to polynomial coefficients.
problem Understanding the volume of hyperbolic links in 3-manifolds.
method Using Kauffman bracket functions and polynomial invariants, linking volume to polynomial coefficients.
result Coefficients of polynomial provide 2-sided linear bounds on the volume of hyperbolic links.
Menasco showed that a non-split, prime, alternating link that is not a 2-braid is hyperbolic in S3. We prove a similar result for links in closed thickened surfaces S×I. We define a link to be fully alternating if it has an alternating projection from S×I to S where the interior of every complemen…
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…
Paper introduces an invariant for knots in non-orientable manifolds, akin to Turaev's comultiplication.
problem Defining an invariant for pseudo-classical knots in non-orientable thickening of a non-orientable surface.
method Introducing an invariant Δ that is an analogue of Turaev comultiplication, defined in terms of homotopy classes of loops on the surface. result Analogous invariants to affine index polynomial for pseudo-classical knots in non-orientable manifolds.
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.
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.
We use the Bar-Natan Zh-correspondence to identify the generalized Alexander polynomial of a virtual knot with the Alexander polynomial of a two component welded link. We show that the Zh-map is functorial under concordance, and also that Satoh's Tube map (from welded links to ribbon knotted tori in S4) is functoria…
New theorem bounds link volume using surface coefficients.
problem Bounding hyperbolic volume of links on surfaces.
method Analogue of Dasbach-Lin theorem for surface links.
result Bounds on link volume from surface polynomial coefficients.