Paper generalizes Yamada polynomial to virtual spatial graphs.
problem Generalizing classical knot theory to virtual spatial graphs.
method Topological definition and combinatorial approach for virtual spatial graphs.
result Generalized Yamada polynomial defined and proven invariant.
The Kauffman-Vogel polynomials are three variable polynomial invariants of 4-valent rigid vertex graphs. A one-variable specialization of the Kauffman-Vogel polynomials for unoriented 4-valent rigid vertex graphs was given by using the Kauffman bracket and the Jones-Wenzl idempotent colored with 2. Bataineh, Elha…
The notion of a pseudoknot is defined as an equivalence class of knot diagrams that may be missing some crossing information. We provide here a topological invariant schema for pseudoknots and their relatives, 4-valent rigid vertex spatial graphs and singular knots, that is obtained by replacing unknown crossings or ve…
New proof for global rigidity of vertex scaling on polyhedral surfaces.
problem Global rigidity of vertex scaling on polyhedral surfaces.
method Elementary variational proof based on continuity of eigenvalues and extension of convex functions.
result Global rigidity of vertex scaling proved without involving 3D hyperbolic geometry.
In \cite{4} Kauffman and Vogel constructed a rigid vertex regular isotopy invariant for unoriented four-valent graphs embedded in three dimensional space. It assigns to each embedded graph G a polynomial, denoted [G], in three variables, A, B and a, satisfies the skein relation: $$ [\psdiag{2}{6}{overcross}]=…
Study uses knot theory to model RNA foldings, emphasizing both entanglement and intrachain interactions.
problem Modeling RNA foldings considering both entanglement and intrachain interactions.
method Combines knot theory with embedded rigid vertex graphs to emphasize both entanglement and intrachain interactions of RNA foldings.
result Defines and computes a coloring counting invariant for stuck links, providing explicit computations for arc diagrams of RNA foldings.
Graphically discrete groups have strong rigidity properties.
problem Understanding the rigidity of group actions on graphs.
method Introducing graphical discreteness and proving rigidity properties.
result Free products of graphically discrete groups are action rigid.
The paper studies graph products of groups and recovers graph and vertex groups under certain conditions.
problem Recovering graph and vertex groups from graph products of groups.
method Using non-generic almost positive sentences, the authors show that under specific conditions, the underlying graph and vertex groups can be recovered.
result The core of the defining graph determines an invariant of the elementary theory of a right-angled Artin group.
We construct bundles of modules of vertex operator algebras, and prove the rigidity and vanishing theorem for the Dirac operator on loop space twisted by such bundles. This result generalizes many previous results.
The main motivation here is a question: whether any polyhedron which can be subdivided into convex pieces without adding a vertex, and which has the same vertices as a convex polyhedron, is infinitesimally rigid. We prove that it is indeed the case for two classes of polyhedra: those obtained from a convex polyhedron b…
In this paper we prove that the space of flat metrics (nonpositively curved Euclidean cone metrics) on a closed, oriented surface is marked length spectrally rigid. In other words, two flat metrics assigning the same lengths to all closed curves differ by an isometry isotopic to the identity. The novel proof suggests a…
Proves rigidity of circle packings in the plane, generalizing previous work.
problem Rigidity of infinite inversive distance circle packings in the plane.
method Maximal principle for generic weighted Delaunay inversive distance circle packings and ring lemma for inversive distance circle packings in hexagonal triangulated plane.
result Proves Bowers-Stephenson's conjecture for inversive distance circle packings.
Let P be a (non necessarily convex) embedded polyhedron in R3, with its vertices on an ellipsoid. Suppose that the interior of P can be decomposed into convex polytopes without adding any vertex. Then P is infinitesimally rigid. More generally, let P be a polyhedron bounding a domain which is the union of p…
Study finds non-isotopic transverse tori in Engel manifolds.
problem Identifying distinct transverse tori in Engel manifolds.
method Constructing an infinite family of non-isotopic transverse tori, introducing a homological invariant.
result Found an infinite family of non-isotopic transverse tori that are smoothly isotopic.
Graphs of surface groups are quasi-isometrically rigid.
problem Understanding when graphs of surface groups are quasi-isometrically equivalent.
method Analyzing the quasi-isometric rigidity of graphs of surface groups with a cyclic JSJ decomposition.
result Any group quasi-isometric to a graph of surface groups is abstractly commensurable with it.
The study examines the realizability of a 4-manifold invariant for homeomorphisms.
problem Realizing the Casson-Sullivan invariant for homeomorphisms of 4-manifolds.
method Investigation of the invariant's realizability and application to surface isotopy.
result The invariant can be realized fully after stabilizing with a single S2imesS2 for all orientable pairs of homeomorphic 4-manifolds. New invariants distinguish spatial graphs not previously possible.
problem Distinguishing spatial graphs using Dehn colorings.
method Developed vertex-weight invariants based on Dehn colorings.
result Found spatial graphs distinguishable by vertex-weight invariants.
Graph Lie algebras have infinite prolongation if they have a vertex of degree one.
problem Determining when the prolongation of a graph Lie algebra is infinite-dimensional.
method Analyzing labeled direct graphs and their associated Lie algebras.
result Graph Lie algebras are infinite-dimensional if and only if they have a vertex of degree one.
Explicit presentations found for asymptotically rigid mapping class groups.
problem Understanding the structure of asymptotically rigid mapping class groups.
method Using a graph of groups structure, we compute explicit presentations.
result Computed explicit presentations for asymptotically rigid mapping class groups of surfaces.
Proves minimal crossing diagrams for specific spatial graphs.
problem Proving minimal crossing diagrams for spatial graphs.
method Analyzing adequate diagrams and replacing vertices and edges.
result All 1-vertex spatial graphs with adequate diagrams have minimal crossing number.
New gauge theory invariant detects non-smooth isotopy of P2-knots.
problem Detecting non-smooth isotopy of P2-knots. method Real Seiberg-Witten theory to construct a gauge theoretic invariant.
result Found a family of P2-knots that are topologically isotopic but not smoothly isotopic. Paper proves rigidity of discrete conformal structures on polyhedral surfaces.
problem Rigidity of discrete conformal structures on polyhedral surfaces.
method Variational principles.
result Proves Glickenstein's conjecture on the rigidity of discrete conformal structures.
Groups acting on product trees are boundary rigid.
problem Understanding boundary rigidity of groups acting on product trees.
method Analyzing geometric actions and visual boundaries of groups.
result Visual boundaries of CAT(0) spaces are homeomorphic to a join of two Cantor sets.
Anosov surfaces with same length spectrum are isometric.
problem Identifying metrics on surfaces based on their length spectrum.
method Combining microlocal tools with complex curve geometry.
result Metrics with the same length spectrum on Anosov surfaces are isometric.
Study exotic knottings of surfaces in 4-manifolds via symmetries.
problem Understanding knotted surfaces in 4-manifolds and their symmetries.
method Developed a recipe for projectively rigid surfaces and used techniques from convex geometry and hyperbolic geometry.
result Found finer knottedness phenomena through successive knotting, revealing more complex symmetries.
Marked vertex diagrams provide a combinatorial way to represent knotted surfaces in R4; including virtual crossings allows for a theory of virtual knotted surfaces and virtual cobordisms. Biquandle counting invariants are defined only for marked vertex diagrams representing knotted orientable surfaces; we e…
We considered a surgery, called Lagrangian attaching disk surgery, that can be applied to a Lagrangian surface L at the presence of a Lagrangian attaching disk D, to obtain a new Lagrangian surface L' which is always smoothly isotopic to L. We showed that this type of surgery includes all even generalized Dehn twists a…
We use microlocal sheaf theory to show that if two knots have Legendrian isotopic conormal tori, then the knots are isotopic or mirror images.
Study on deformation of affine structures on Lie groups using cohomology.
problem Understanding the gap between global topological invariants and left-invariant affine structures.
method Comparing De Rham and Koszul-Vinberg cohomology groups on Lie groups SO(2), H3(R), and SGal(3).
result Vanishing theorem for the second KV-cohomology group, proving structural rigidity of orbits.
Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. Fleming and the author introduced some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the constituent 2-component algebraically split li…
New method to classify simple Smale flows on S3.
problem Classifying simple Smale flows on S3. method Embedded template and Kauffman's invariant of spatial graphs.
result Isotopic classification of simple Smale flows on S3. Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. We introduce some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the 2-component constituent algebraically split links and show examples…
New method uses mosaics to study wild knots.
problem Classifying wild knots with infinite knotting behavior.
method Extending knot mosaic theory to represent wild knots with isolated wild points.
result Developed a framework for mosaic tangles and mosaic rigid vertex spatial graphs.
Given an m-component link L in S3 (m≥2), we construct a family of links which are link homotopic, but not link isotopic, to L. Every proper sublink of such a link is link isotopic to the corresponding sublink of L. Moreover, if L is an unlink then there exist links that in addition to the above prope…
In this note we show that compact self shrinkers in R3 are "topologically standard" in that any genus g compact self shrinker is ambiently isotopic to the standard genus g embedded surface in R3. As a consequence self shrinking tori are unknotted.
Finite type invariants separate PL links in 3D space.
problem If two PL links are isotopic, are they PL isotopic?
method Proved PL isotopic links are indistinguishable by finite type invariants.
result Finite type invariants separate PL links in 3D space if and only if certain conjectures hold.
Techniques of gauge theory are used to define and compute an invariant of certain diffeomorphisms of 4-manifolds. The invariant vanishes for any diffeomorphism which is smoothly isotopic to the identity. As an application, we give the first example of a diffeomorphism of a simply-connected 4-manifold which is homotopic…
Characters from logarithmic VOAs linked to torus link invariants.
problem Understanding characters of logarithmic vertex operator algebras.
method Relating characters to coloured Jones invariants of torus links.
result Characters of logarithmic VOAs are limits of coloured Jones invariants of torus links.
The paper studies circle packings on surfaces with boundary and their total geodesic curvatures.
problem Existence and rigidity of circle packings with conical singularities.
method Variational principle and combinatorial Ricci flow.
result Existence and rigidity of circle packings with prescribed total geodesic curvature.
New invariant shows Dehn twist on connected sum of homology tori is not isotopic to identity.
problem Determining when Dehn twists on connected sums of homology tori are isotopic to identity.
method Generalized Pin(2)-equivariant family Bauer-Furuta invariant to nonsimply connected manifolds and constructed a refinement.
result Dehn twist on X1#X2 is not isotopic to identity if determinants r1,r2 are odd. Researchers clarify modular group representations and vertex operator algebras for 3d invariants.
problem Understanding the full set of 3d invariants and their modular properties.
method Introducing supersymmetric defects and constructing cone vertex operator algebras.
result The full vector-valued quantum modular form for \(\widetilde{
m SL}_2(\mathbb{Z})\) captures all \(\hat Z\)-invariants of a given three-manifold.
Study rigidity and volume optimization of hyperbolic polyhedra.
problem Rigidity and volume optimization of hyperbolic polyhedra.
method Analyzing decorated 1-3 type hyperbolic polyhedra and their metrics.
result Decorated 1-3 type hyperbolic polyhedra are rigid up to isometry and change of decorations.
Study of bonded knots and braids with new algebraic models.
problem Classifying and understanding physical or chemical bonds in knots and braids.
method Developed new algebraic models (bonded knots, braids, braidoids) and invariants.
result New algebraic structures and invariants for bonded knots and braids.
We introduce the multiplexing of a crossing, replacing a classical crossing of a virtual link diagram with multiple crossings which is a mixture of classical and virtual. For integers mi (i=1,…,n) and an ordered n-component virtual link diagram D, a new virtual link diagram D(m1,…,mn) is ob…
Study graph products of groups, classifying them up to measure equivalence and rigidity.
problem Classifying graph products of groups up to measure equivalence and rigidity.
method Measure-theoretic and structural properties of von Neumann algebras, rigidity theorems.
result Quantified measure equivalence classification and rigidity theorems for graph products.
Computes link invariants in real projective 3-space using topological vertex.
problem Computing link invariants in RP3. method Uses geometric transition and toric Calabi-Yau threefold.
result Findings are series in Kahler parameters with positivity property.
Smooth orbit equivalence proves metric equivalence for geodesic flows.
problem Proving metric equivalence for geodesic flows under orbit equivalence.
method Proving metric equivalence for geodesic flows under orbit equivalence.
result Smooth orbit equivalence implies conformal equivalence of metrics.
New proof associates partitions to isotopic pseudo-Anosov homeomorphisms.
problem Stable and unstable foliations for pseudo-Anosov homeomorphisms.
method Geometric realization of Fathi's result for isotopic homeomorphisms.
result Associated stable and unstable partitions for isotopic pseudo-Anosov homeomorphisms.