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.
Study of Penner's cocycle on fatgraph complex.
We introduce an invariant for trivalent fatgraph spines of a once bordered surface, which takes values in the first homology of the surface. This invariant is the secondary object coming from two 1-cocycles on the dual fatgraph complex, one introduced by Morita and Penner in 2008, and the other by Penner, Turaev, and t…
We prove that every trivalent marked bordered fatgraph comes equipped with a canonical generalized Magnus expansion in the sense of Kawazumi. This Magnus expansion is used to give canonical lifts of the higher Johnson homomorphisms , for , to the Torelli groupoid, and we provide a recursive combinatorial …
We show that Galois conjugates of stretch factors of pseudo-Anosov mapping classes arising from Penner's construction lie off the unit circle. As a consequence, we show that for all but a few exceptional surfaces, there are examples of pseudo-Anosov mapping classes so that no power of them arises from Penner's construc…
Penner coordinates are extended to the Teichmüller spaces of oriented closed surfaces.
For all orientable closed surfaces, we determine the minimal dilatation among mapping classes arising from Penner's construction. We also discuss generalisations to surfaces with punctures.
For any nonorientable closed surface, we determine the minimal dilatation among pseudo-Anosov mapping classes arising from Penner's construction. We deduce that the sequence of minimal Penner dilatations has exactly two accumulation points, in contrast to the case of orientable surfaces where there is only one accumula…
We give a new description of the Arnoux-Yoccoz mapping classes as a product of two Dehn twists and a finite order element. The construction is analogous to Penner's construction of mapping classes with small stretch factors.
We produce a one-parameter family of coordinates of the decorated Teichmüller space of an ideally triangulated punctured surface with negative Euler characteristic, which is a deformation of Penner's simplicial coordinate \cite{P1}. If , the decorated Teichmüller space in…
In this paper, we study the Galois conjugates of stretch factors of pseudo-Anosov elements of the mapping class group of a surface. We show that - except in low-complexity cases - these conjugates are dense in the complex plane. For this, we use Penner's construction of pseudo-Anosov mapping classes. As a consequence, …
We extend the canonical cell decomposition due to Epstein and Penner of a hyperbolic manifold with cusps to the strictly convex setting. It follows that a sufficiently small deformation of the holonomy of a finite volume strictly convex real projective manifold is the holonomy of some nearby projective structure with r…
Generalising a seminal result of Epstein and Penner for cusped hyperbolic manifolds, Cooper and Long showed that each decorated strictly convex projective cusped manifold has a canonical cell decomposition. Penner used the former result to describe a natural cell decomposition of decorated Teichmüller space of puncture…
Geometric proof shows primes of form 3k+1 are norms of Eisenstein integers.
Let M be the moduli space of irreducible flat PSL(2,R) connections on a punctured surface of finite type with parabolic holonomies around punctures. By using a notion of admissibility of an ideal arc, M is covered by dense open subsets associated to ideal triangulations of the surface. A principal bundle over M is cons…
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.
Shows Anosov flows with genus one sections, supporting a conjecture.
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…
New tiling algorithm for hyperbolic 3-manifolds, characterizing cusp areas.
We prove that the dilatation of any pseudo-Anosov homeomorphism on a translation surface that belong to a hyperelliptic component is bounded from below uniformly by sqrt{2}. This is in contrast to Penner's asymptotic. Penner proved that the logarithm of the least dilatation of any pseudo-Anosov homeomorphism on a surfa…
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…
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.
Study of quandle coloring quivers with dihedral quandles.
New invariant for spin 3-manifolds using super 3-cocycles.
This paper connects spinors to horospheres in hyperbolic space.
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…
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.
We describe all the situations in which the Kontsevich-Zorich cocycle has zero Lyapunov exponents. Confirming a conjecture of Forni, Matheus, and Zorich, this only occurs when the cocycle satisfies additional geometric constraints. We also describe the real Lie groups which can appear in the monodromy of the Kontsevich…