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.
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.
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.
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 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 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.
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.
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.
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…
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 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.
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.
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 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. 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.
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…
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 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. 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.
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.
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.
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…
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.
New invariants detect non-slice knots in thickened surfaces.
problem Detecting non-slice knots in thickened surfaces.
method Adapting Milnor's theory to virtual knots and using Bar-Natan's Zh map.
result New invariants obstruct concordance to homologically trivial knots.
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.
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…
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.
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.
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. 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.
Several authors have recently studied virtual knots and links because they admit invariants arising from R-matrices. We prove that every virtual link is uniquely represented by a link L in S X I, a thickened, compact, oriented surface S, such that the link complement (S X I) - L has no essential vertical cylinder.
Essential surfaces in link diagrams on surfaces are crucial for understanding link properties.
problem Understanding the essentiality of surfaces in link diagrams on surfaces.
method Proving checkerboard surfaces are π1-essential and contain no essential closed curves that are ∂-parallel.
result Checkerboard surfaces of alternating link diagrams are π1-essential and contain no essential closed curves that are ∂-parallel.
Twisted links are a generalization of virtual links. As virtual links correspond to abstract links on orientable surfaces, twisted links correspond to abstract links on (possibly non-orientable) surfaces. In this paper, we introduce the notion of the double covering of a twisted link. It is defined by considering the o…
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…
Characterizes arithmetic and commensurable links in curved surfaces.
problem Classifying arithmetic and commensurable links in curved surfaces.
method Combines symmetry arguments, combinatorial geometry, and number-theoretic data.
result Characterizes arithmetic and commensurable right-angled tiling links.
New quantum invariants for 3-manifolds with boundary calculated.
problem Quantum invariants of compact 3-manifolds with non-vacuous boundary.
method Virtual link diagrams and surgery on manifolds of the form FimesI. result Explicit calculations of invariants for some examples.
A polynomial invariant of virtual links, arising from an invariant of links in thickened surfaces introduced by Jaeger, Kauffman, and Saleur, is defined and its properties are investigated. Examples are given that the invariant can detect chirality and even non-invertibility of virtual knots and links. Furthermore, it …
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.
Two link diagrams on compact surfaces are strongly equivalent if they are related by Reidemeister moves and orientation preserving homeomorphisms of the surfaces. They are stably equivalent if they are related by the two previous operations and adding or removing handles. Turaev and Turner constructed a link homology f…
A virtual link is a generalization of a classical link that is defined as an equivalence class of certain diagrams, called virtual link diagrams. It is further generalized to a twisted link. Twisted links are in one-to-one correspondence with stable equivalence classes of links in oriented thickenings of (possibly non-…
The paper studies right-angled links on higher genus surfaces.
problem Classifying and understanding right-angled links on surfaces of higher genus.
method Defining and proving equivalence of properties for RGCR links, using diagram restrictions and polygonal checkerboard surfaces.
result Classification of RGCR links and bounds on their number for a given genus.
New homology invariant for links in surfaces, a deformation of APS.
problem Defining a new homology invariant for links in surfaces.
method A 1-parameter family of homology invariants, motivated by instanton Floer homology.
result The new invariant recovers APS homology and has a stronger detection property.
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…
We extend the theory of hyperbolicity of links in the 3-sphere to tg-hyperbolicity of virtual links, using the fact that the theory of virtual links can be translated into the theory of links living in closed orientable thickened surfaces. When the boundary surfaces are taken to be totally geodesic, we obtain a tg-hype…