A cumbersome hypothesis for Viro patchworking of real algebraic curves is the convexity of the given subdivision. It is an open question in general to know whether the convexity is necessary. In the case of trigonal curves we interpret Viro method in terms of dessins d'enfants. Gluing the dessins d'enfants in a coheren…
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48 results for “Viro's patchworking”
Transversal special parabolic points in the graph of a polynomial obtained under Viro's patchworkingmath.AG
The study builds families of polynomials with a specific number of special parabolic points.
problem Finding polynomials with a given number of special parabolic points.
method Viro's patchworking theorem to construct families of polynomials.
result Built a family of degree d polynomials with special parabolic points.
New method uses patchworking to represent oriented matroids.
problem Representing oriented matroids from polyhedral structures.
method Variant of Viro's patchworking, controlled cell merging process.
result Achieved a combinatorial manifestation of patchworking.
Classifies real algebraic curves on a quadric ellipsoid of specific degree.
problem Classifying real algebraic curves of bidegree (5,5) on the quadric ellipsoid.
method Reduction to curves in the second Hirzebruch surface, combining classical construction methods on toric surfaces.
result Previously known restrictions form a complete system for this bidegree.
New method for constructing real algebraic surfaces from complex tropical hypersurfaces.
problem Constructing real algebraic surfaces from complex tropical hypersurfaces.
method Non-Abelian patchworking theory.
result The method reproduces all existing isotopy types of surfaces up to degree three.
A bit of tropical geometrymath.AG
This friendly introduction to tropical geometry is meant to be accessible to first year students in mathematics. The topics discussed here are basic tropical algebra, tropical plane curves, some tropical intersections, and Viro's patchworking. Each definition is explained with concrete examples and illustrations. To a …
This text is intended to become in the long run Chapter 3 of our long saga dedicated to Riemann, Ahlfors and Rohlin. Yet, as its contents evolved as mostly independent (due to our inaptitude to interconnect both trends as strongly as we wished), it seemed preferable to publish it separately. More factually, our account…
New Alexandrov-Patchwork construction for Lorentzian spaces with curvature bounds.
problem Understanding finite diameter constraints in Lorentzian geometry.
method Constructing Alexandrov-Patchwork and proving Bonnet-Myers theorem for Lorentzian spaces.
result Lorentzian spaces with curvature bounds have finite diameter.
Cluster Quilting clusters fragmented data sets for neuroscience and genomics.
problem Clustering fragmented data sets in neuroscience and genomics.
method Cluster Quilting method using patch ordering, patchwise SVD, sequential linear mapping, and k-means.
result Cluster Quilting discovers more accurate clusters than other methods.
Paper proposes a copula method to generate unfavorable VaR scenarios.
problem Creating unfavorable VaR scenarios for insurance models.
method Patchwork copulas to create unfavorable VaR scenarios with given marginal distributions.
result Demonstrated with a 19-dimensional real-life insurance losses data set.
The divergence theorem in its usual form applies only to suitably smooth vector fields. For vector fields which are merely piecewise smooth, as is natural at a boundary between regions with different physical properties, one must patch together the divergence theorem applied separately in each region. We give an elegan…
We show that for every spherical category $\C$ with invertible dimension, the Turaev-Viro TQFT admits a splitting into blocks which come from an HQFT, called the Turaev-Viro HQFT. The Turaev-Viro HQFT has the classifying space $B\grad$ as target space, where $\grad$ is a group obtained from the category $\C$. This cons…
Paper finds Hempel pairs distinguishable by Turaev-Viro invariants.
problem Surface bundles with non-isomorphic fundamental groups.
method Examines Hempel pairs and their Turaev-Viro invariants.
result Examples of Hempel pairs distinguishable and undistinguishable by Turaev-Viro invariants.
In this paper, we examine Kitaev's lattice model for an arbitrary complex, semisimple Hopf algebra. We prove that this model gives the same topological invariants as Turaev-Viro theory. Using the description of Turaev-Viro theory as an extended TQFT, we prove that the excited states of the Kitaev model correspond to Tu…
Turaev-Viro invariants match for certain surface bundles.
problem Matching Turaev-Viro invariants for surface bundles.
method Profinite completions and spherical fusion categories.
result Invariants coincide for specific surface groups.
The paper finds formulas for a specific invariant order 7.
problem Finding a specific invariant order 7.
method Explicit formulas for a Turaev-Viro invariant of order 7.
result Explicit formulas for the invariant of order 7.
Study Turaev-Viro invariants of Seifert fibered 3-manifolds, proving volume conjecture.
problem Understanding the asymptotic behavior of Turaev-Viro invariants for Seifert fibered 3-manifolds.
method Analysis of large asymptotic behavior of Turaev-Viro invariants.
result Proved the volume conjecture for Seifert fibered 3-manifolds with empty and non-empty boundaries.
The paper proves an asymptotic additivity of Turaev-Viro invariants for a family of 3-manifolds.
problem Preserving the Turaev-Viro invariant volume conjecture under gluings of toroidal boundary components.
method Using a construction of hyperbolic cusped 3-manifolds by Agol, the authors show that the asymptotics of Turaev-Viro invariants are additive under certain gluings of elementary pieces.
result The Turaev-Viro invariant volume conjecture is preserved under specific gluings of 3-manifolds.
A relative version of the Turaev-Viro invariants and the volume of hyperbolic polyhedral 3-manifoldsmath.GT
Defines a new version of Turaev-Viro invariants for 3-manifolds with boundaries.
problem Computing the volume of hyperbolic polyhedral 3-manifolds.
method Introduces a relative version of Turaev-Viro invariants for ideally triangulated compact 3-manifolds with boundaries and a coloring on edges.
result Proves the Volume Conjecture for these invariants, suggesting a method to solve the conjecture for hyperbolic 3-manifolds with totally geodesic boundary.
Paper uses Turaev-Viro TQFT to estimate 3-manifold genus.
problem Estimating the Heegaard genus of 3-manifolds.
method Turaev-Viro state sum TQFT and unitary modular category.
result Provides a lower bound for Heegaard genus using TQFT.
Study how Turaev-Viro invariants change with cabling operations.
problem Understanding how Turaev-Viro invariants vary with cabling operations.
method Utilized the invertibility of a linear operator associated with torus knot cable spaces in Reshetikhin-Turaev SO3 TQFT.
result Showed the Chen-Yang volume conjecture is stable under (p,q)-cabling for coprime p and q.
Paper proves stability of volume conjecture under cabling.
problem Stability of volume conjecture under cabling.
method Analyzes Turaev-Viro invariants and simplicial volume.
result Stability of Chen-Yang conjecture under -cabling.
Study Turaev-Viro invariants of 3-manifolds with toroidal boundary.
problem Volume conjecture for Seifert fibered and graph 3-manifolds.
method Large asymptotic behavior of Turaev-Viro invariants under gluing operation.
result Volume conjecture proven for Seifert fibered and graph 3-manifolds.
Formula connects Turaev-Viro invariants to colored Jones polynomials and hyperbolic volumes.
problem Relating Turaev-Viro invariants to hyperbolic volumes of link complements.
method Using colored Jones polynomials and asymptotic behavior of Turaev-Viro invariants.
result Asymptotics of Turaev-Viro invariants determine hyperbolic volumes of specific link complements.
The paper characterizes boundaries in Turaev-Viro TQFTs and Dijkgraaf-Witten theories.
problem Characterizing boundaries in Turaev-Viro TQFTs and Dijkgraaf-Witten theories.
method Identifying explicit boundary locality conditions and proving consistency with state sum models.
result Turaev-Viro and Dijkgraaf-Witten theories with boundary defects admit a state sum description.
We work in the reduced SU(N,K) modular category as constructed recently by Blanchet. We define spin type and cohomological refinements of the Turaev-Viro invariants of closed oriented 3-manifolds and give a formula relating them to Blanchet's invariants. Roberts' definition of the Turaev-Viro state sum is exploited. Fu…
Paper reviews Viro's definition of Khovanov homology for tangles.
problem Defining Khovanov homology for unoriented framed tangles.
method Explains Viro's definition and the relation between -matrix and Kauffman bracket.
result Clarifies the relation between -matrix and Kauffman bracket in Khovanov homology.
We consider certain invariants of links in 3-manifolds, obtained by a specialization of the Turaev-Viro invariants of 3-manifolds, that we call colored Turaev-Viro invariants. Their construction is based on a presentation of a pair (M,L), where M is a closed oriented 3-manifold and L is an oriented link in M, by a tria…
In a previous work arXiv:0903.4512, we have built an homotopical Turaev-Viro invariant and an HQFT from the universal graduation of a spherical category. In the present paper, we show that every graduation of a spherical category $\C$ defines an homotopical Turaev-Viro invariant $HTV_{\C}^{(G,p)}$ and an HQFT $…
In this article, we discuss a (2+1)-dimensional topological quantum field theory, for short TQFT, with a Verlinde basis. As a conclusion of this general theory, we have a Dehn surgery formula. We show that Turaev-Viro-Ocneanu TQFT has a Verlinde basis. Several applications of this theorem are exposed. Based on Izumi's …
New construction of Turaev-Viro invariants invariant under Morita equivalence.
problem Constructing Turaev-Viro invariants invariant under Morita equivalence.
method Pivotal bicategory construction of spherical module categories.
result The invariant recovers the standard Turaev-Viro invariant and is independent of the skeleton.
The paper connects Turaev-Viro invariants and Gromov norm of 3-manifolds.
problem Relating Turaev-Viro invariants and Gromov norm of 3-manifolds.
method Combines TQFT techniques, geometric decomposition theory, and analytical estimates of -symbols.
result Established a relation between the asymptotics of Turaev-Viro invariants and Gromov norm of 3-manifolds.
Asymptotic Behavior of Colored Jones polynomial and Turaev-Viro Invariant of figure eight knotmath.GT
Study on asymptotic behavior of knot invariants for figure eight knot.
problem Investigate asymptotic behavior of colored Jones polynomials and Turaev-Viro invariants for figure eight knot.
method Considered -th colored Jones polynomials and Turaev-Viro invariants for figure eight knot with fixed limiting ratio of and . Found asymptotic expansion formula for colored Jones polynomials and showed exponential growth rate difference for close to 1/2 and 1. Related Turaev-Viro invariants to colored Jones polynomials.
result Asymptotic expansion formula for colored Jones polynomials and Turaev-Viro invariants of figure eight knot.
New invariants from quantum group theory for hyperbolic 3-manifolds.
problem Computing invariants for hyperbolic 3-manifolds with boundary.
method Using modular doubles of quantum and -symbols.
result Invariants decay exponentially with hyperbolic volume and 1-loop terms.
Paper connects quantum representations to fibered links and Turaev-Viro invariants.
problem Quantum representations of surface mapping class groups and their relation to fibered links.
method Relates quantum representations to Turaev-Viro invariants of hyperbolic 3-manifolds.
result Showed exponential growth of Turaev-Viro invariants for fibered links implies AMU conjecture.
Turaev Viro invariants are amongst the most powerful tools to distinguish 3-manifolds: They are implemented in mathematical software, and allow practical computations. The invariants can be computed purely combinatorially by enumerating colourings on the edges of a triangulation T. These edge colourings can be interpre…
In this paper, it is shown that there are no nonconstant Goussarov-Polyak-Viro finite-type invariants that are invariant under the virtualization move. As an immediate corollary, we obtain the theorem which states none of the Birman coefficients of the Jones-Kauffman polynomial are of GPV finite type.
The Turaev-Viro invariants are a powerful family of topological invariants for distinguishing between different 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them require exponential time. The invariants are parameterised by an integer . We resolve the question …
In this paper, we establish a rigorous correspondence between the two tube algebras, that one comes from the Turaev-Viro-Ocneanu TQFT introduced by Ocneanu and another comes from the sector theory introduced by Izumi, and construct a canonical isomorphism between the centers of the two tube algebras, which is a conjuga…
The article calculates asymptotic expansions for quantum invariants from surgeries on Whitehead link components.
problem Calculating quantum invariants for 3-manifolds resulting from surgeries on Whitehead link components.
method Asymptotic expansion of relative Reshetikhin-Turaev and Turaev-Viro invariants.
result Asymptotic formulas for both invariants are derived.
We describe the Polyak-Viro arrow diagram formulas for the coefficients of the Conway polynomial. As a consequence, we obtain the Conway polynomial as a state sum over some subsets of the crossings of the knot diagram. It turns out to be a simplification of a special case of Jaeger's state model for the HOMFLY polynomi…
By applying a variant of the TQFT constructed by Blanchet, Habegger, Masbaum, and Vogel, and using a construction of Ohtsuki, we define a module endomorphism for each knot K by using a tangle obtained from a surgery presentation of K. We show that it is strong shift equivalent to the Turaev-Viro endomorphism associated…
We introduce a class of combinatorial hypersurfaces in the complex projective space. They are submanifolds of codimension~2 in $\C P^n$ and are topologically "glued" out of algebraic hypersurfaces in $(\C^*)^n$. Our construction can be viewed as a version of the Viro gluing theorem, relating topology of algebraic hyper…
In this paper we show how one can extend Turaev-Viro invariants, defined for an arbitrary spherical fusion category , to 3-manifolds with corners. We demonstrate that this gives an extended TQFT which conjecturally coincides with the Reshetikhin-Turaev TQFT corresponding to the Drinfeld center . In the present…
Paper provides seven Gauss diagram formulas for degree three long virtual knots.
problem Tackles the complete list of seven distinct Gauss diagram formulas for degree three long virtual knots.
method Gives seven Gauss diagram formulas for degree three long virtual knots and 23 for classical knots.
result Each Gauss diagram formula for degree three long virtual knots is represented as classical knots formulas, supporting Goussarov-Polyak-Viro conjecture.
New inequality for odd-degree flexible curves using surface doubling.
problem Bounding the number of non-empty ovals of odd-degree flexible curves.
method Defining an Arnold surface for odd-degree flexible curves and using it to derive a Viro--Zvonilov-type inequality.
result Upper bound on the number of non-empty ovals of odd-degree flexible curves.
Proves volume conjectures for figure-eight knot surgeries.
problem Volume conjectures for hyperbolic 3-manifolds.
method Ohtsuki's method applied to figure-eight knot surgeries.
result Proves Asymptotic Expansion and Volume Conjectures for figure-eight knot surgeries.
We present a new method to produce simple formulas for 1-cocycles of knots over the integers, inspired by Polyak-Viro's formulas for finite-type knot invariants. We conjecture that these formulas always represent finite-type cohomology classes in the sense of Vassiliev. An example of degree 3 is studied, and shown to c…