Paper investigates pluripotential theory on Teichmüller space using new methods.
problem Understanding pluripotential theory on Teichmüller space.
method Alternative approach to Krushkal formula, natural stratified structure, Levi form description.
result Natural stratified structure and description of Levi form.
Recently V. Krushkal and D. Renardy generalized the Tutte polynomial from graphs to cell complexes. We show that evaluating this polynomial at the origin gives the number of cellular spanning trees in the sense of A. Duval, C. Klivans, and J. Martin. Moreover, after a slight modification, the Tutte-Krushkal-Renardy pol…
Formulae for 1-3 handle attachments in 4-manifolds.
problem Understanding handle attachments in 4-dimensional manifolds.
method Developed general formulae for 1-3 handle attachments using skein modules.
result Derived complete description of gluing homomorphism on skein modules.
New invariant connects knot homology and BPS series for plumbed knot complements.
problem Understanding invariants of plumbed knot complements.
method Introducing an invariant unifying knot lattice homology and BPS series, proving a surgery formula.
result Proved a surgery formula relating the new invariant to the weighted graded root of the surgered 3-manifold.
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.
In this paper, we show that the extremal length functions on Teichmüller space are log-plurisubharmonic. As a corollary, we obtain an alternative proof of L.Liu and W.Su's results on the plurisubharmonicity of extremal length functions. We also obtain alternative proofs of S.Krushkal's results that a function defined b…
New links in 3-manifolds have large systole.
problem Geometric constraints on hyperbolic links in 3-manifolds.
method Constructing hyperbolic filling links with large essential systole.
result Links have large essential systole, contrasting earlier constraints.
Paper proves triple linking form vanishes under specific conditions.
problem Analyzing rational homology cobordism and linking forms.
method Proves vanishing of triple torsion linking form under specific conditions.
result Triple torsion linking form vanishes on a specific Lagrangian.
Proves a theorem for surface links, simplifying diagrams of links.
problem Understanding minimal diagrams of surface links.
method Two-variable generalization of Jones polynomial for surface links.
result Reduced alternating diagrams of surface links have minimal crossing number.
New compacta with unique embedding properties found.
problem Embedding properties of compacta in high dimensions.
method Using sticky Cantor sets and sequences of compacta.
result Construction of compacta with specific embedding properties.
Affirmative proof that rank 3 3-manifolds have filling links.
problem Existence of filling links in rank 3 3-manifolds.
method Introduced and used the concept of filling links, proved for rank 3 manifolds.
result Every rank 3 closed orientable 3-manifold contains a filling link.
We discuss a new perspective on Khovanov homology, using categorifications of tensor products. While in many ways more technically demanding than Khovanov's approach (and its extension by Bar-Natan), this has distinct advantage of directly connecting Khovanov homology to a categorification of \$(\mathbb{C}^2)^{\otimes …
We introduce a graphical calculus for computing morphism spaces between the categorified spin networks of Cooper and Krushkal. The calculus, phrased in terms of planar compositions of categorified Jones-Wenzl projectors and their duals, is then used to study the module structure of spin networks over the colored unknot…
We extend Lipshitz-Sarkar's definition of a stable homotopy type associated to a link L whose cohomology recovers the Khovanov cohomology of L. Given an assignment c (called a coloring) of positive integer to each component of a link L, we define a stable homotopy type X_col(L_c) whose cohomology recovers the c-colored…
Unified theories for colored sl(2) knot homology.
problem Equivalence of different models for colored sl(2) knot homology.
method Conceptualized properties into a Chebyshev system and proved its uniqueness.
result Equivalence of Khovanov and Cooper-Krushkal models for the unknot.
We present the Round Handle Problem, proposed by Freedman and Krushkal. It asks whether a collection of links, which contains the Generalised Borromean Rings, are slice in a 4-manifold R constructed from adding round handles to the four ball. A negative answer would contradict the union of the surgery conjecture and th…
Clarifies sign ambiguity in Khovanov homology functoriality.
problem Sign ambiguity in Khovanov homology functoriality.
method Blanchet's oriented model and cobordisms with singularities.
result Strict functoriality established for the oriented model.
For a graph embedded into a surface, we relate many combinatorial parameters of the cycle matroid of the graph and the bond matroid of the dual graph with the topological parameters of the embedding. This will give an expression of the polynomial, defined by M.Las Vergnas in a combinatorial way using matroids as a spec…
New findings show infinitely many knots cannot be smoothly round handle slices.
problem Understanding the smoothability of knots in 4-dimensional space.
method Analyzing the properties of knots under surgery conjectures and cobordism conjectures.
result Infinitely many knots fail to be smoothly round handle slices.
Bounding twist number of surface links using polynomial coefficients.
problem Bounding the twist number of alternating surface links.
method Introducing a generalized homological Kauffman bracket and applying it to surface link diagrams.
result A bound for the twist number of alternating surface links in terms of polynomial coefficients.
Identifies a mod-p triple cup product for rational homology 3-spheres with specific first homology.
problem Locally flat embeddings in S4 for rational homology 3-spheres method Using triple torsion linking form and torsion-linking duality
result Identifies the mod-p triple cup product for specific rational homology 3-spheres Ropelength and embedding thickness are related measures of geometric complexity of classical knots and links in Euclidean space. In their recent work, Freedman and Krushkal posed a question regarding lower bounds for embedding thickness of n-component links in terms of the Milnor linking numbers. The main goal of the…
Extends quantum annular homology to infinite sets.
problem Quantum annular homology and its applications.
method Extension of Burnside categories to infinite sets and application to quantum annular Khovanov spectrum.
result Quantum annular Khovanov spectrum with infinite cyclic group action.
New embeddings show answer to Baker-Laidacker question can be yes or no.
problem Answer to Baker-Laidacker question about disjoint compacta in R^N.
method Use of specific wild Cantor sets and Antoine's methods.
result Answer to Baker-Laidacker question can be twofold.
The paper shows examples of 2-complexes that can't be embedded in R^4, hiding obstructions in higher Milnor invariants.
problem Embedding 2-complexes in R^4 with hidden obstructions.
method Provides examples of 2-complexes and families of PL immersions that hide embedding obstructions.
result Embedding obstructions vanish for the given examples, answering a question in Avramidi-Okun-Schreve's paper.
Jones-Wenzl projectors lifted to Khovanov spectra, proving knot conjectures.
problem Understanding Jones-Wenzl projectors in Khovanov spectra.
method Constructing and studying lifted projectors via maps and polynomial actions.
result Complete computation of 3-colored Khovanov spectrum of the unknot, proving conjectures.
2D complexes can be almost-embedded in 4D space without self-intersections.
problem Understanding the embedding properties of 2-dimensional complexes in 4-dimensional space.
method Analyzing specific 2-complexes constructed by Freedman-Krushkal-Teichner and showing they can be PL immersed in R4 without self-intersections. result Many 2-complexes can be PL almost-embedded in R4 with singularities only as self-intersections of some 2-cells. We construct an action of a polynomial ring on the colored sl(2) link homology of Cooper-Krushkal, over which this homology is finitely generated. We define a new, related link homology which is finite dimensional, extends to tangles, and categorifies a scalar-multiple of the sl(2) Reshetikhin-Turaev invariant. We expe…
Short note observes quantum Hochschild homology as a composition of known operations.
problem Quantum Hochschild homology as a new invariant of annular links.
method Observes quantum Hochschild homology as a composition of two known operations.
result Quantum Hochschild homology is a valid invariant of annular links.
New stable homotopy refinement of quantum annular Khovanov homology.
problem Quantum topological Hochschild homology and annular Khovanov spectra.
method Introducing quantum topological Hochschild homology (qTHH) and constructing a new stable homotopy refinement of quantum annular Khovanov homology.
result The new stable homotopy refinement agrees with qTHH of spectral Chen-Khovanov tangle bimodules and recovers earlier work.
Study of Milnor invariants and ropelength of spherical links.
problem Understanding the relationship between the thickness of spherical links and their Milnor invariants.
method Generalized Massey products and Milnor invariants to spherical links, finding optimal asymptotic bounds.
result Optimal asymptotic bounds on Milnor invariants in terms of thickness, revealing a polynomial vs exponential regime.
The validity of Freedman's disk theorem is known to depend only on the fundamental group. It was conjectured that it fails for nonabelian free fundamental groups. If this were true then surgery theory would work in dimension four. Recently, Krushkal and Lee proved a surprising result that surgery theory works for a lar…
We generalize categorified Jones-Wenzl projectors in odd Khovanov homology.
problem Categorify Jones-Wenzl projectors for odd Khovanov homology.
method Develop grading multicategories to replace grading categories, proving existence and uniqueness of categorified projectors.
result Existence and uniqueness of categorified Jones-Wenzl projectors in odd Khovanov homology.
We introduce and study the notion of the G-Tutte polynomial for a list A of elements in a finitely generated abelian group Γ and an abelian group G, which is defined by counting the number of homomorphisms from associated finite abelian groups to G. The G-Tutte polynomial is a common generalizatio…
Classifies flat knots up to 8 crossings using Lyndon words.
problem Classifying flat knots up to a certain number of crossings.
method Using matchings on Lyndon words and various knot invariants.
result Distinguished all flat knots up to 7 crossings except for five pairs.
Researchers calculate (t,q)-series invariants for Seifert manifolds.
problem Calculating (t,q)-series invariants for Seifert manifolds. method Generalization of earlier work on plumbed 3-manifolds to Seifert manifolds with b1=0. result Calculation of (t,q)-series invariants for Seifert manifolds with b1=0. Infinite families of quantum modular invariants for 3-manifolds are discovered.
problem Quantum modular forms and invariants of 3-manifolds.
method Lattice cohomology and BPS q-series of 3-manifolds, recently developed theory by AJK. result First calculation of AJK series for an infinite family of 3-manifolds.
Let EMBED(k,d) be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into R^d? Known results easily imply polynomiality of EMBED(k,2) (k=1,2; the case k=1, d=2 is graph planarity) and of EMBED(k,2k) for all k>2 (even if k i…
Study TMF-valued TQFT for closed 3-manifolds using torsion linking pairings.
problem Explicit description of TMF-module state space for closed 3-manifolds.
method Construct canonical invariants and tokenization from torsion linking pairings.
result Explicit model for TMF-module state space in terms of rank-one TMF-module.
A clear understanding of topology of higher-dimensional objects is important in many branches of both pure and applied mathematics. In this survey we attempt to present some results of higher-dimensional topology in a way which makes clear the visual and intuitive part of the constructions and the arguments. In particu…
New non-rigid discrete groups found in hyperbolic spaces.
problem Uniqueness of conformal or spherical CR structures on spheres.
method Nilpotent Sierpiński carpet and stretching to construct non-rigid groups.
result Discrete hyperbolic groups can have non-rigid deformations.
The paper proves embedding conditions for complexes in manifolds using matrix rank criteria.
problem Embedding k-dimensional simplicial complexes into (k−1)-connected PL manifolds. method Proves embedding conditions using a skew-symmetric matrix with low rank over Q. result Embedding conditions for k-complexes in 2k-manifolds are equivalent to low-rank matrix conditions. Proves a general connected sum formula for families Seiberg-Witten invariants.
problem Limited connected sum formulae for families Seiberg-Witten theory.
method Develops a general connected sum formula incorporating previous results.
result Proves a new connected sum formula for Seiberg-Witten families.
Derives an integral formula for G2-structures.
problem Calculating properties of G2-structures.
method Applies an integral formula for G-structures to G2.
result Derives an integral formula relating curvatures and quadratic invariants.
Derives integral formulae on weighted manifolds.
problem No specific problem stated; focuses on mathematical derivations.
method Introduces weighted mean sigma-r curvature and uses weighted Newton transformations.
result Derives integral formulae generalizing previous work.
Paper derives trace formula for magnetic Laplacian at zero energy.
problem Trace formula for magnetic Laplacian at zero energy.
method Generalizes Gutzwiller trace formula, focuses on zero energy level.
result Derives trace formula at zero energy level.
The Gauss formula is extended to various Laplacians on submanifolds.
problem Deriving formulas for Laplacians on submanifolds.
method Extending the Gauss formula to different types of Laplacians.
result Formulas for various Laplacians on submanifolds.
The paper is devoted to the problem of finding explicit combinatorial formulae for the Pontryagin classes. We discuss two formulae, the classical Gabrielov-Gelfand-Losik formula based on investigation of configuration spaces and the local combinatorial formula obtained by the author in 2004. The latter formula is based…