New operations on Khovanov homology refine knot invariants.
problem Understanding finer knot invariants through homological operations.
method Developed an algebra of homological operations on Khovanov homology and lifted it to integral Khovanov homology.
result Conjectured infinite algebras of homological operations and provided evidence.
We consider various homological operations on homology of quandles. We introduce the notion of quandle partial derivatives, and extreme chains on which appropriate partial derivatives vanish. Extreme chains yield homological operations. We also consider the degree one homology operations created using elements of the q…
K-homology classes linked to elliptic operators.
problem Defining K-homology classes for elliptic operators.
method Using uniform K-homology and principal symbols.
result Classes depend only on the operator's principal symbol.
Two definitions of set-theoretic Yang-Baxter homology are shown to be equivalent.
problem Equivalence of two Yang-Baxter homology definitions.
method Comparison of algebraic and graphic homology theories.
result The graphic homology is equivalent to the algebraic one.
New operator reveals unique features of sl(N) link homology.
problem Understanding unique features of sl(N) link homology.
method Constructing an operator on sl(N) link homology with mod N coefficients.
result Structural features of sl(P) link homology are revealed.
The paper simplifies the computation of a complex polynomial using Yang-Baxter operators.
problem Computing the homology of Yang-Baxter operators for arbitrary m.
method Reduced the computation to initial conditions and produced explicit formulas.
result Explicit formulas for the third and fourth homology.
This paper proves the Atiyah-Singer index theorem for Dirac operators.
problem Proving the Atiyah-Singer index theorem for Dirac operators.
method Presentation of the theorem as a computation of the K-homology of a point.
result Proof of the Atiyah-Singer index theorem for Dirac operators.
This paper defines a unified homology theory for quandles that are unions of groups.
problem Developing a homology theory for quandles that incorporate group operations.
method Defining multiple conjugation quandles and a homology theory that considers both group and quandle operations.
result Characterization of the first homology group and cocycle invariants for handlebody-links.
This paper proves the Atiyah-Singer index theorem for elliptic operators.
problem Proving the Atiyah-Singer index theorem for elliptic operators.
method Computing geometric K-cycles corresponding to analytic K-cycles determined by elliptic operators.
result Proof of the Atiyah-Singer index theorem for elliptic operators.
New operations match Steenrod squares on Khovanov homology.
problem Matching Steenrod squares with operations on Khovanov homology.
method Applied cup-i products to Khovanov functor and proved agreement with Steenrod squares.
result Lipshitz-Sarkar's Sq^2 agrees with Morán's sq^2.
New knots found with non-trivial Steenrod operations on Khovanov homology.
problem Identifying knots with non-trivial Steenrod operations on Khovanov homology.
method Examined prime, hyperbolic, and satellite knots using Steenrod operations.
result Found knots (prime, hyperbolic, satellite) with non-trivial Steenrod operations on Khovanov homology.
Geometric approach simplifies K-homology computation for Lie manifolds.
problem Computing the Fredholm index of fully elliptic operators on Lie manifolds.
method Adapting geometric K-homology concepts, introducing geometric cycles and a comparison map.
result Reduction of index computation to Dirac operator index with a smoothing operator.
Uniform K-homology theory applied to elliptic operators on manifolds with boundary.
problem Developing a theory to study boundary conditions for elliptic operators on non-compact manifolds.
method Theory of relative uniform K-homology, developing a relative index map.
result Uniform K-homology classes of boundary conditions and their connection to the higher ρ-invariant.
New operators in Khovanov-Rozansky homology exhibit symmetry.
problem Symmetry in Khovanov-Rozansky homology.
method Defined new commuting operators Fk and proved F2 satisfies hard Lefschetz property. result Symmetry in Khovanov-Rozansky homology is confirmed.
Incompatible operations affect Khovanov homology and spectral sequences.
problem Incompatibility between operations and spectral sequences.
method Observation of obstructions to integral lifting and spectrification.
result Lipshitz-Sarkar Steenrod operations are incompatible with Szabo's spectral sequence.
Study shows flexibility of homology groups of Reeb spaces of fold maps through surgery operations.
problem Understanding changes in homology groups of Reeb spaces of fold maps.
method Introduced surgery operations (bubbling operations) to fold maps and used elementary theory of sequences and continuous functions.
result Homology groups of Reeb spaces of fold maps constructed by iterations of these operations are flexible and can be represented as direct sums of original homology groups and finitely generated commutative groups.
Survey of computations for Riemann surface moduli spaces, focusing on unstable homology.
problem Computing unstable homology of moduli spaces of Riemann surfaces.
method Integral, mod-2, and rational coefficient computations; use of homology operations.
result Explicit generators of unstable homology for most cases determined.
New 3-manifolds bound rational 4-balls through specific operations.
problem Finding rational homology 3-spheres that bound rational homology 4-balls.
method Two operations that preserve lattice embedding obstruction to bounding rational homology balls.
result Explicit examples of rational surgeries on torus knots that bound rational homology balls.
Using the unbounded picture of analytical K-homology, we associate a well-defined K-homology class to an unbounded symmetric operator satisfying certain mild technical conditions. We also establish an ``addition formula'' for the Dirac operator on the circle and for the Dolbeault operator on closed surfaces. Two proofs…
An odd vector field Q on a supermanifold M is called homological, if Q2=0. The operator of Lie derivative LQ makes the algebra of smooth tensor fields on M into a differential tensor algebra. In this paper, we give a complete classification of certain invariants of homological vector fields called character…
In a previous paper we constructed a spectrum-level refinement of Khovanov homology. This refinement induces stable cohomology operations on Khovanov homology. In this paper we show that these cohomology operations commute with cobordism maps on Khovanov homology. As a consequence we obtain a refinement of Rasmussen's …
Study pinched submanifolds, proving homology vanishing results.
problem Understanding the geometry and topology of pinched submanifolds.
method Investigates submanifolds with a pinching condition on extrinsic invariants.
result Homology vanishing theorems for pinched submanifolds.
Adjusts Yang-Baxter operators for HOMFLYPT polynomials.
problem Computing 2-cocycle invariants for links.
method Adjusts Yang-Baxter operators and computes second homology.
result Potential for 2-cocycle invariant for links.
By studying spaces of flow graphs in a closed oriented manifold, we construct operations on its cohomology, parametrized by the homology of the moduli spaces of compact Riemann surfaces with boundary marked points. We show that the operations satisfy the gluing axiom of an open homological conformal field theory. This …
New operations defined on moduli spaces for bundles with orientations.
problem Pushforward operations for principal bundles with orientations.
method Developed a general theory of pushforward operations for principal G-bundles, constructing specific operations for G=BU(1). result Classified all stable pushforward operations and showed they are generated by the projective Euler and rank operations.
The monodromy action in the homology of level sets of Morse functions on stratified singular analytic varieties is studied. The local variation operators in both the standard and the intersection homology groups defined by the loops around the critical values of such functions are reduced to similar operators in the ho…
We derive a formula for the μˉ-invariant of a Seifert fibered homology sphere in terms of the eta-invariant of its Dirac operator. As a consequence, we obtain a vanishing result for the index of certain Dirac operators on plumbed 4-manifolds bounding such spheres.
Study initiates homology theory for Bol-Moufang quasigroups.
problem Developing a homology theory for a specific type of quasigroups.
method Using extensions by affine quasigroups, define boundary operations and compute homology groups.
result Computed second homology groups for examples, speculating on relation to functorial homology.
Formula for satellite operators using knot Floer homology.
problem Computing knot Floer homology for satellite knots.
method Using Heegaard Floer Dehn surgery formulas and formal knot Floer complexes.
result Formulas to compute knot Floer homology for satellite knots.
This paper reinterprets Khovanov-Sano symmetries using BV formalism.
problem Understanding symmetries in equivariant Khovanov homology.
method Identifying Shumakovitch operator as a BV Laplacian and proving L∞-algebra structure. result Construction of an intrinsic L∞-algebra on the Khovanov-Sano complex. Formula for Heegaard Floer multicurves of double tangles from knot complements.
problem Computing Heegaard Floer multicurve invariants of double tangles.
method Using a simple formula derived from knot complements, comparing with Khovanov homology.
result New characterisation of L-space knots and first example of thin knot Floer homology.
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.
Chas and Sullivan have defined an intersection-type product on the homology of the free loop space LM of an oriented manifold M. In this paper we show how to extend this construction to a topological conformal field theory of degree d. In particular, we get operations on the homology of LM which are parameterized by th…
We study the Hochschild homology groups of the algebra of complete symbols on a foliated manifold (M,F). The first step is to relate these groups to the Poisson homology of (M,F) and of other related foliated manifolds. We then establish several general properties of the Poisson homology groups of foliated manifold…
We generalize Roe's Index Theorem for operators of Dirac type on open manifolds to elliptic pseudodifferential operators. To this end we introduce a class of pseudodifferential operators on manifolds of bounded geometry which is more general than similar classes defined by other authors. We revisit Spakula's uniform K-…
In a previous paper, we defined a space-level version X(L) of Khovanov homology. This induces an action of the Steenrod algebra on Khovanov homology. In this paper, we describe the first interesting operation, Sq^2:Kh^{i,j}(L) -> Kh^{i+2,j}(L). We compute this operation for all links up to 11 crossings; this, in turn, …
Characterizes homology d-manifolds with g2=3 for d≥3.
problem Classify homology d-manifolds with specific g2 values.
method Combinatorial characterization and operations like joins, retriangulations, and connected sums.
result Homology d-manifolds with g2=3 are spheres and can be derived from previous ones.
New operations on loop space chains from string diagrams.
problem Understanding operations on loop space chains.
method Constructing string diagrams and operations on their chains.
result Recovering known loop space homology structure.
A new graph classification method using persistent homology.
problem Graph classification with graph connectivity structure.
method Learnable filter function for persistent homology computation.
result Empirically, the method compares favorably to previous techniques.
We generalize Roe's index theorem for graded generalized Dirac operators on amenable manifolds to multigraded elliptic uniform pseudodifferential operators. This generalization will follow as a corollary from a local index theorem that is valid on any manifold of bounded geometry. This local formula incorporates the un…
Topological gauge theories in four dimensions which admit surface operators provide a natural framework for realizing homological knot invariants. Every such theory leads to an action of the braid group on branes on the corresponding moduli space. This action plays a key role in the construction of homological knot inv…
We generalize the very well known boundary operator of the ordinary singular homology theory, defined in many books about algebraic topology. We describe a variant of this ordinary simplicial boundary operator where the usual boundary (n-1)-simplices of each n-simplex are replaced by combinations of internal (n-1)- sim…
The paper explores slice disks and their properties using satellite operations and knot Floer homology.
problem Tackles the conjecture about slice disks and their positive Whitehead doubles.
method Uses techniques from knot Floer homology, Seiberg-Witten theory, and Khovanov homology.
result Provides evidence for the conjecture and constructs exotic disks.
New link detection results using closures of 3-braids.
problem Link detection using homology theories.
method Closure operations on 3-braids and homology theories.
result Detection of specific links using link Floer homology, Khovanov homology, and annular Khovanov homology.
The paper explores knot theory from Fox colorings to Yang-Baxter homology.
problem Developing new invariants for knot theory.
method Generalizing Fox colorings to racks and quandles, then to Yang-Baxter operators and categorifying the Jones polynomial.
result Building homology of Yang-Baxter operators and speculating on co-cycle invariants.
New method for calculating loop operations on surfaces.
problem Computing algebraic operations on loops in surfaces.
method Using fillings of surfaces by graphs to compute homological intersection number, Lie bracket, and Lie cobracket.
result Effective new approach to standard algebraic operations on loops.
Lipshitz and Sarkar recently introduced a space-level refinement of Khovanov homology. This refinement induces a Steenrod square operation $\Sq^2$ on Khovanov homology which they describe explicitly. This paper presents some computations of $\Sq^2$. In particular, we give examples of links with identical integral Khova…
Analyzes K-homology classes of singular complex spaces.
problem Analyzing K-homology classes of singular complex spaces.
method Examines various L2-∂ complexes and their rolled-up operators. result Analytic K-homology classes of singular spaces can be related to their birational invariants.