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.
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48 results for “string diagrams”
In previous work, the author defined the intersection graph of a chord diagram associated with string links (as in the theory of finite type invariants). In this paper, we classify the trees which can be obtained as intersection graphs of string link diagrams.
A virtual string can be defined as an equivalence class of planar diagrams under certain kinds of diagrammatic moves. Virtual strings are related to virtual knots in that a simple operation on a virtual knot diagram produces a diagram for a virtual string. In this paper we consider three operations on a virtual string …
Study knot diagrams on a sphere without vertical lines, focusing on minimal crossings.
problem Understanding minimal crossings of knot diagrams on a punctured sphere.
method Mathematical model of string figures using knot diagrams on -space with missing vertical lines, analyzing minimal crossings under Reidemeister moves.
result Minimal number of crossings of knot diagrams on a punctured sphere.
New presentation of Goussarov-Habiro Lie algebra using primitive Feynman diagrams.
problem Defining a filtration of string links using clasper surgeries and geometrically realizing Feynman diagrams.
method Concrete presentation of the rational Goussarov-Habiro Lie algebra using primitive Feynman diagrams and relations.
result Alternative diagrammatic proof of Massuyeau's rational version of the Goussarov-Habiro conjecture.
In previous work, we defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we look at the case when this graph is a tree, and we show that in many cases these trees determine the chord diagram (modulo the usual 1-term and 4-term rela…
Multistring based matricesmath.GT
A new method for virtual homotopy classes of virtual strings.
problem Addressing an open problem of Turaev regarding virtual homotopy classes of virtual strings.
method Generalizing Turaev's method to multistring based matrices for virtual n-strings.
result Construction of similar invariants for virtual homotopy classes of virtual n-strings.
Developing tools for computing string amplitudes with hyperbolic vertices.
problem Computing off-shell string amplitudes with new vertices.
method Constructing local coordinates and investigating limits for hyperbolic three-string vertex.
result Derived conservation laws and performed sample computations.
String vertices proven in hyperbolic geometry, unique up to transformations.
problem Existence and uniqueness of string vertices in string field theory.
method Homological proof using hyperbolic metrics and geodesic boundaries.
result String vertices are sets of surfaces with systole greater than or equal to .
New formulas for spatial 2-bouquet graphs discovered.
problem Finding formulas for Vassiliev invariants of spatial 2-bouquet graphs.
method Introducing new Gauss diagram formulas for flat vertex isotopy classes of spatial 2-bouquet graphs.
result First simple example of a Gauss diagram formula for spatial 2-bouquet graphs.
Study of spaces of pure braids and string links using diagrams and integrals.
problem Understanding spaces of pure braids and string links through algebraic structures.
method Use of Kontsevich's CDGA of diagrams and Chen's iterated integrals to establish Hopf algebra isomorphisms and connections.
result Established a correspondence between Milnor invariants and Chen integrals for Brunnian spherical links.
The Reshetikhin-Turaev invariant, Turaev's TQFT, and many related constructions rely on the encoding of certain tangles (n-string links, or ribbon n-handles) as n-forms on the coend of a ribbon category. We introduce the monoidal category of Hopf diagrams, and describe a universal encoding of ribbon string links as Hop…
Using intersection theory in the context of Hilbert manifolds and geometric homology we show how to recover the main operations of string topology built by M. Chas and D. Sullivan. We also study and build an action of the homology of reduced Sullivan's chord diagrams on the singular homology of free loop spaces, extend…
Study configuration space integrals for Milnor invariants of string links and trivalent trees.
problem Understanding Milnor invariants of string links using combinatorial methods.
method Combinatorial analysis of trivalent homotopy link diagrams and configuration space integrals.
result Established a correspondence between Milnor invariants and linear combinations of trivalent trees.
Proves colored HOMFLYPT polynomial is -holonomic.
problem Proving the -holonomic property of colored HOMFLYPT polynomials.
method Skew Howe duality and Poincare-Birkhoff-Witt computation of quantum group.
result Existence of super-polynomial for knots.
Develops a calculus for knotted objects, including classical links.
problem Characterizing and classifying knotted objects, including classical links.
method Introduces Arrow presentations and w-tree presentations to encode knot diagrams, uses clasper theory for welded knots.
result Characterizes finite type invariants of welded knots and long knots.
Quantum equations for knotsmath.GT
Paper introduces equations to distinguish knots without using knot invariants.
problem Distinguishing knots using only topological and combinatorial methods.
method Linear systems of equations derived from HOMFLYPT and Kauffman polynomials.
result First examples of distinguishing knots without knot invariants.
We prove all knots can be transformed into a trefoil using special diagrams.
problem Transforming any knot into a trefoil using magic tricks.
method Introducing knotholder diagrams to encode transformations.
result All knots can be transformed into a trefoil.
Study on homotopy type of Sullivan diagrams space.
problem Homotopy type of Sullivan diagrams space.
method Vanishing and non-vanishing results, connectivity increase with components, genus stabilization maps, fundamental group computation, homology class construction.
result Non-trivial fundamental group and homology classes of harmonic compactification.
New method encodes knots using clasp diagrams for easier study of invariants.
problem Encoding and studying knots and links more efficiently.
method Introducing clasp diagrams and equivalence relations on them.
result Explicit formula for Alexander-Conway polynomial and Vassiliev invariants.
In this work, we discuss graph like image of curves under moment maps and their relation with the Newton polygon of the curve, which has applications to Lagrangian torus fibration of Calabi-Yau manifolds.
Parity on based matricesmath.GT
Parity defined for based matrices, a new example of virtual knot parity.
problem Defining parity for a new algebraic structure.
method Introduced parity for based matrices, defined reduced stable parity.
result New example of parity for virtual knots.
The second author previously discussed how classical complexity separation conjectures, we call them "axioms", have implications in three manifold topology: polynomial length stings of operations which preserve certain Jones polynomial evaluations cannot produce exponential simplifications of link diagrams. In this pap…
Configuration space integrals have in recent years been used for studying the cohomology of spaces of (string) knots and links in for since they provide a map from a certain differential algebra of diagrams to the deRham complex of differential forms on the spaces of knots and links. We refine this…
Study on coloring virtual tangles with integer and modular arithmetic.
problem Characterizing Fox colorings of virtual tangle diagrams.
method Analyzed classical and virtual tangle diagrams using vector representations and divisibility conditions.
result For , realizability depends on divisibility of the alternating sum. For , all vectors are realizable.
We discuss an action of the Grothendieck-Teichmüller proalgebraic group on the linear span of proalgebraic tangles, oriented tangles completed by a filtration of Vassiliev. The action yields a motivic structure on tangles. We derive distinguished properties of the action particularly on proalgebraic string links and on…
Study shows knots from certain braids are algebraically slice under specific conditions.
problem Characterizing knots formed by specific braids.
method Gauss diagram formulas to analyze algebraic sliceness.
result Knots from are algebraically slice if and only if is odd.
We present ADHM-Nahm data for instantons on the Taub-NUT space and encode these data in terms of Bow Diagrams. We study the moduli spaces of the instantons and present these spaces as finite hyperkahler quotients. As an example, we find an explicit expression for the metric on the moduli space of one SU(2) instanton. W…
Study compares different types of knotted objects and their equivalence relations.
problem Comparing usual, virtual, and welded knotted objects.
method Using Gauss diagram formulae, the study examines self-crossing changes and self-virtualizations.
result Differences between usual, virtual, and welded knotted objects are highlighted.
A model of random walk on knot diagrams is used to study the Alexander polynomial and the colored Jones polynomial of knots. In this context, the inverse of the Alexander polynomial of a knot plays the role of an Ihara-Selberg zeta function of a directed weighted graph, counting with weights cycles of random walk on a …
Virtual knots arise in the study of Gauss diagrams and Vassiliev invariants of usual knots. Virtual braids correspond naturally to virtual knots. We consider the group of virtual braids on n strings VB_n and its Burau representation, in particular we study their homological properties. We prove that the plus-constructi…
We study the string topology of a closed oriented Riemannian manifold M. We describe a compact moduli space of diagrams, and show how the cellular chain complex of this space gives algebraic operations on the singular chains of the free loop space LM of M. These operations are well-defined on the homology of a quotient…
A general state-sum construction of 2-dimensional topological quantum field theories with defectsmath.QA
This research extends TQFTs to include defects using state-sum construction.
problem Extending topological quantum field theories to include defects.
method General state-sum construction with extended Pachner moves, requiring flag-likeness of triangulations.
result Derivation of equations and string diagrams for 2D TQFTs with defects.
The classical Hurwitz numbers of degree n together with the Hurwitz numbers of the seamed surfaces of degree n give rise to the Klein topological field theory. We extend this construction to the Hurwitz numbers of all degrees at once. The corresponding Cardy-Frobenius algebra is induced by arbitrary Young diagrams and …
The paper explores extensions of local moves on string links and their relation to ribbon surfaces.
problem Finding a unique extension of classical local moves on string links.
method Relating local moves to surgeries on ribbon surfaces in R^4.
result There can be at most one unique welded extension of a classical move that is a ribbon residue.
Bar-Natan used Chinese characters to show that finite type invariants classify string links up to homotopy. In this paper, I construct the correct spaces of chord diagrams and Chinese characters for links up to homotopy. I use these spaces to show that the only rational finite type invariants of link homotopy are the p…
This reports on the fundamental objects revealed by Ross Street, which he called `orientals'. Street's work was in part inspired by Robert's attempts to use N-category ideas to construct nets of C*-algebras in Minkowski space for applications to relativistic quantum field theory: Roberts' additional challenge was that …
New algebra defined for Legendrian submanifolds, preserving key invariants.
problem Defining a new algebra to preserve invariants of Legendrian submanifolds.
method Combining string topology techniques with combinatorial methods to count holomorphic disks.
result The new algebra is a filtered, differential graded algebra that captures invariants of Legendrian submanifolds.
We use Polyak's skein relation to give a new proof that Milnor's string link homotopy invariants are finite type invariants, and to develop a recursive relation for their associated weight systems. We show that the obstruction to the triviality of these weight systems is the presence of a certain kind of spanning tree …
This paper connects geometric diagrams to spherical T-duality.
problem Establishing equivalence between geometric and string-theory frameworks.
method Introducing -diagrams and proving their equivalence to spherical T-dual pairs.
result Concrete geometric realization of spherical T-duality.
Large Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transportstat.ML
New framework for efficient PD averaging and clustering.
problem Challenges in averaging and clustering persistence diagrams.
method Reformulate PD metrics as optimal transport problems, leveraging recent computational advances.
result Scalable computations of PD barycenters and clustering on thousands of diagrams.
Motivated by the algebraic open-closed string models, we introduce and discuss an infinite-dimensional counterpart of the open-closed Hurwitz theory describing branching coverings generated both by the compact oriented surfaces and by the foam surfaces. We manifestly construct the corresponding infinite-dimensional equ…
We continue the study of the genus of knot diagrams, deriving a new description of generators using Hirasawa's algorithm. This description leads to good estimates on the maximal number of crossings of generators and allows us to complete their classification for knots of genus 4. As applications of the genus 4 classifi…
New knot polynomials distinguish knot orientations without using knot groups.
problem Distinguishing knots based on their orientations without relying on knot groups.
method Constructing combinatorial 1-cocycles on moduli spaces of knots and cables, using Gauss diagram formulas and local parameterization.
result Polynomial invariants that can distinguish knot orientations.
Researchers find minimal-area metrics on surfaces with constraints.
problem Finding the conformal metric of least area on surfaces with length conditions.
method Numerical convex programs to solve the minimal-area problem.
result The metric is positively curved in some regions and flat in others.
New knot invariant detects non-invertibility, distinguishing all knots.
problem Detecting non-invertibility of knots.
method Refined Vassiliev invariant for 2-cables of framed long knots.
result Invariant distinguishes non-invertible knots.
Knot theory applied to proteins, distinguishing folded linear chains.
problem Classifying proteins as unknots when intra-chain interactions are ignored.
method Developing knot theory for folded linear molecular chains, considering self-bonding, and using Gauss codes and quandles.
result Extended knot theory to distinguish topologies of proteins with intra-chain bonds.
Derives path integrals for perturbative strings on various backgrounds.
problem Calculating path integrals for strings on curved backgrounds.
method Derives path integrals from string geometry theory by considering fluctuations around string backgrounds.
result Derives path integrals of all order perturbative strings on various backgrounds.