A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Refined 1-cocycle for knots helps quantify isotopies.
The construction of characteristic classes via the curvature form of a connection is one motivation for the refinement of integral cohomology by de Rham cocycles -- known as differential cohomology. We will discuss the analog in the case of a group action on the manifold: The definition of equivariant characteristic fo…
New invariant for spin 3-manifolds using super 3-cocycles.
By introducing a refinement of the Goldman-Turaev Lie bialgebra, we interpret the divergence cocycle in the Kashiwara-Vergne problem and the Enomoto-Satoh obstructions for the surjectivity of the Johnson homomorphisms as some part of a regular homotopy version of the Turaev cobracket.
We present a cocycle model for elliptic cohomology with complex coefficients in which methods from 2-dimensional quantum field theory can be used to rigorously construct cocycles. For example, quantizing a theory of vector bundle-valued fermions yields a cocycle representative of the elliptic Thom class. This construct…
What are called secondary characteristic classes in Chern-Weil theory are a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology. We consider the problem of refining the construction of secondary characteristic classes from cohomology sets to cocycle spaces; and …
Holomorphic functions from knot complements link to quantum modular forms.
We formulate differential cohomology and Chern-Weil theory -- the theory of connections on fiber bundles and of gauge fields -- abstractly in the context of a certain class of higher toposes that we call "cohesive". Cocycles in this differential cohomology classify higher principal bundles equipped with cohesive struct…
This paper establishes a correspondence between biquandle and quandle colorings for classical and surface links.
Develops Morse homology with DG coefficients for manifolds and spaces.
Constructs a model for differential KO-theory using Clifford modules.
The higher gauge field in 11-dimensional supergravity -- the C-field -- is constrained by quantum effects to be a cocycle in some twisted version of differential cohomology. We argue that it should indeed be a cocycle in a certain twisted nonabelian differential cohomology. We give a simple and natural characterization…
New group-theoretic Johnson classes applied to curves with torsion Ceresa classes.
In the background effective field theory of heterotic string theory, the Green-Schwarz anomaly cancellation mechanism plays a key role. Here we reinterpret it and its magnetic dual version in terms of differential twisted String- and differential twisted Fivebrane-structures that generalize the notion of Spin-structure…
We introduce new polynomial isotopy invariants for closed braids. They are constructed as polynomial valued {\em Gauss diagram 1-cocycles} evaluated on the full rotation of the closed braid around the core of the corresponding solid torus. They can be calculated with polynomial complexity with respect to the b…
Develops a new index theory for odd Z/kZ K-theory.
In this paper we study perturbations of constant cocycles for actions of higher rank semi-simple algebraic groups and their lattices. Roughly speaking, for ergodic actions, Zimmer's cocycle superrigidity theorems implies that the perturbed cocycle is measurably conjugate to a constant cocycle modulo a compact valued co…
Study projective derivative cocycles for circle diffeomorphisms.
T. Mochizuki determined all 3-cocycles of the third quandle cohomologies of Alexander quandles on finite fields. We show that all the 3-cocycles, except those of 2-cocycle forms, are derived from group 3-cocycles of a meta-abelian group. Further, the quandle cocycle invariant of a link using Mochizuki's 3-cocycle is eq…
Virtual index cocycles reformulate virtual link invariants.
We give a construction of quandle cocycles from group cocycles, especially, for any integer p \geq 3, quandle cocycles of the dihedral quandle R_p from group cocycles of the cyclic group Z/p. We will show that a group 3-cocycle of Z/p gives rise to a non-trivial quandle 3-cocycle of R_p. When p is an odd prime, since d…
Study on group cocycles for volume-preserving diffeomorphisms.
The paper extends Euler class theory to measurable cocycles.
We derive general Novikov-Morse type inequalities in a Conley type framework for flows carrying cocycles, therefore generalizing our results in [FJ2] derived for integral cocycle. The condition of carrying a cocycle expresses the nontriviality of integrals of that cocycle on flow lines. Gradient-like flows are distingu…
Differential Cohomotopy theory predicts brane interactions via chord diagrams.
New rack and multiple group rack cohomology for surfaces in 3-sphere.
New algebraic rules for 5D shapes based on 3D cocycles.
Researchers compute a residue cocycle for Dirac-type operators using modified Getzler calculus.
We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure. These invariants coincide with the classical Yang-Baxter cocycle invariants for classical knots but provide extra information about virt…
The theory of quandle (co)homology and cocycle knot invariants is rapidly being developed. We begin with a summary of these recent advances. One such advance is the notion of a dynamical cocycle. We show how dynamical cocycles can be used to color knotted surfaces that are obtained from classical knots by twist-spinnin…
We consider cocycles of isometries on spaces of nonpositive curvature . We show that the supremum of the drift over all invariant ergodic probability measures equals the infimum of the displacements of continuous sections under the cocycle dynamics. In particular, if a cocycle has uniform sublinear drift, then there…
New shifting chain map enhances quandle invariants for links.
Quantum cocycle invariants derived from Yang-Baxter cohomology.
We introduce the notion of pullback along a measurable cocycle and we use it to extend the Borel invariant studied by Bucher, Burger and Iozzi to the world of measurable cocycles. The Borel invariant is constant along cohomology classes and has bounded absolute value. This allows to define maximal cocycles. We conclude…
Develops Patterson-Sullivan theory for coarse cocycles.
We incorporate quandle cocycle information into the quandle coloring quivers we defined in arXiv:1807.10465 to define weighted directed graph-valued invariants of oriented links we call \textit{quandle cocycle quivers}. This construction turns the quandle cocycle invariant into a small category, yielding a categorifica…
We explore a knot invariant derived from colorings of corresponding -tangles with arbitrary connected quandles. When the quandle is an abelian extension of a certain type the invariant is equivalent to the quandle -cocycle invariant. We construct many such abelian extensions using generalized Alexander quandles w…
Enhances psyquandle counting invariants using cocycles.
Paper extends multiplicative constants to measurable cocycles theory.
Study of Penner's cocycle on fatgraph complex.
Study of quandle coloring quivers with dihedral quandles.
Paper studies knotoid chirality using shadow quandle colorings and invariants.
Quandle 2-cocycles define invariants of classical and virtual knots, and extensions of quandles. We show that the quandle 2-cocycle invariant with respect to a non-trivial -cocycle is constant, or takes some other restricted form, for classical knots when the corresponding extensions satisfy certain algebraic condit…
We refine the cyclic cohomological apparatus for computing the Hopf cyclic cohomology of the Hopf algebras associated to infinite primitive Cartan-Lie pseudogroups, and for the transfer of their characteristic classes to foliations. The main novel feature is the precise identification as a Hopf cyclic complex of the im…
Study parametrized Kähler class for cocycles on Hermitian symmetric spaces.
We propose a simple method to produce quandle cocycles from group cocycles, as a modification of Inoue-Kabaya chain map. We further show that, in respect to "universal central extended quandles", the chain map induces an isomorphism between their third homologies. For example, all Mochizuki's quandle 3-cocycles are sho…
The state-sum invariants for knots and knotted surfaces defined from quandle cocycles are described using the Kronecker product between cycles represented by colored knot diagrams and a cocycle of a finite quandle used to color the diagram. Such an interpretation is applied to evaluating the invariants. Algebraic inter…
This paper defines maximal measurable cocycles for surface groups into Hermitian Lie groups and studies their algebraic hulls.