Geometric deep learning predicts knot invariants.
problem Predicting knot invariants from knot data.
method Constructing a functor from knots to graphs and using graph neural networks.
result High generalization capabilities demonstrated.
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168,742 papers · 148 categories
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48 results for “geometric knot theory”
This paper uses sheaf theory to model virtual knots geometrically.
problem Defining and understanding virtual knots in a geometric framework.
method Sheaf theory applied to virtual knot diagrams to model them geometrically.
result A geometric model for virtual knots formalizes the intuitive notion of knots in a variable ambient space.
Hyperbolic Knot Theorymath.GT
Book introduces hyperbolic geometry for knot theory.
problem Understanding knots through hyperbolic geometry.
method Explains hyperbolic geometry, geometric structures, and techniques.
result Develops three knot invariants from hyperbolic geometry.
Survey of various non-classical knot theories from geometric and algebraic perspectives.
problem Various modifications to classical knot theory.
method Comparative geometric and algebraic analysis of non-classical knot theories.
result Distinct topological and combinatorial features in generalized knot theories.
Geometric duality connects graph isomorphism and knot equivalence.
problem Understanding the equivalence of graph isomorphism and knot equivalence.
method Observation of geometric duality in planar graphs and links.
result The equivalence relation defined by isomorphisms of checkerboard graphs is the same as 2-isomorphisms of checkerboard graphs.
The paper explores how topological methods can reveal insights into electric charge distributions on knots.
problem Understanding the qualitative behavior of electric potentials on knots.
method Geometric topology techniques applied to electrostatics.
result Proved a lower bound on the size of the critical set based on knot projections.
This paper develops a new homology theory for biquandles and discusses geometric realizations.
problem Constructing knot invariants using set-theoretic Yang-Baxter equation.
method Developed a normalized (co)homology theory for biquandles and geometrically realized them.
result Geometric realization of biquandles has finitely generated second homotopy group for finite biquandles.
Simplified geometric derivation of quantum A-polynomials for knots.
problem Deriving quantum A-polynomials for knots in a simple geometric way.
method Geometric derivation using Ward identities in Chern-Simons theory, contact geometry, and Kauffman calculus.
result Simplified presentation of quantum A-polynomials, making them accessible to a broader audience.
The Palais-Smale condition is proven for various knot energies.
problem Existence and smoothness of minimizing knots in geometric knot theory.
method Proof of the Palais-Smale condition for specific knot energies.
result Existence of minimizing knots and long-time existence of their flows.
Highly twisted knots can be geometrically triangulated.
problem Proving geometric triangulations for hyperbolic 3-manifolds.
method Using highly twisted knots and extending Gueritaud and Schleimer's work.
result Sufficiently highly twisted knots admit a geometric triangulation.
Plumbers' knotsmath.AT
We construct a new type of geometric knot theory, plumbers' knots, and solve the problems of distinguishing and enumerating such knots at a fixed level of complexity. (v2) Minor edits, added theorem 3.18. (v3) Substantial revisions, essentially completely rewritten in places.
We present an invariant of a three-dimensional manifold with a framed knot in it based on the Reidemeister torsion of an acyclic complex of Euclidean geometric origin. To show its nontriviality, we calculate the invariant for some framed (un)knots in lens spaces. Our invariant is related to a finite-dimensional fermion…
This is survey about the classical knot concordance group, prepared for an upcoming handbook of knot theory. Topics include: the basic definitions of concordance; the theory of algebraic concordance as developed by Levine; the theory of Casson-Gordon invariants; applications of topological surgery as developed by Freed…
The paper classifies knots in real projective 3-space and introduces new geometric tools.
problem Classifying knots in real projective 3-space and understanding their properties.
method Structural theorem, space bending surgery, genus definition, non-cancellation theorem.
result The genus detects knottedness and classifies knots in real projective 3-space.
Geometric finiteness theory for essential surfaces in knot exteriors with geometric bounds.
problem Understanding the topology of essential surfaces in knot exteriors with geometric constraints.
method Developed a relative geometric finiteness theory using bounded geometry and thickness conditions.
result Every bounded-geometry slice contains only finitely many pair-isotopy classes, and topology is recoverable from finite geometric data.
Sutured manifolds defined by Gabai are useful in the geometrical study of knots and 3-dimensional manifolds. On the other hand, homology cylinders are in an important position in the recent theory of homology cobordisms of surfaces and finite-type invariants. We study a relationship between them by focusing on sutured …
Study SO(3)-knot states for torus complements, linking to simplicial volume.
problem Capturing simplicial volume of knot complements via knot states.
method Holomorphic sections, geometric quantization, Witten-Chern-Simons theory.
result Large r asymptotics of -norm of SO(3)-knot states relate to simplicial volume.
Geometric representations of cycles in quandle homology theory are given in terms of colored knot diagrams. Abstract knot diagrams are generalized to diagrams with exceptional points which, when colored, correspond to degenerate cycles. Bounding chains are realized, and used to obtain equivalence moves for homologous c…
Survey of knot polynomials and their categorification, including quiver-knot correspondence.
problem Understanding the relationship between knot polynomials and quivers.
method Overview of classical knot polynomials, physical and geometric insights, and 3d theory analysis.
result Exploration of the LMOV invariants and their connection to BPS states.
Note on spherical quandlesmath.GT
The paper connects knot representations and spherical quandle colorings.
problem Defining the Casson-Lin invariant for knots.
method One-to-one correspondence between -representations and spherical quandle colorings.
result Geometric interpretation of the trace-free condition for the Casson-Lin invariant.
We explain new developments in classical knot theory in 3 and 4~dimensions, i.e. we study knots in 3-space, up to isotopy as well as up to concordance. In dimension~3 we give a geometric interpretation of the Kontsevich integral (joint with Jim Conant), and in dimension 4 we introduce new concordance invariants using v…
The recently conjectured knots-quivers correspondence relates gauge theoretic invariants of a knot in the 3-sphere to representation theory of a quiver associated to the knot. In this paper we provide geometric and physical contexts for this conjecture within the framework of the large duality of Ooguri…
We construct many examples of non-slice knots in 3-space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3-dimensional topological knot concordance group. The bottom part of the filtration exhibits all…
Study convex embeddability in linear and circular orders, applying to knots.
problem Understanding the quasi-order of convex embeddability in linear and circular orders.
method Combinatorial and descriptive set-theoretic methods applied to arcs and knots.
result Established combinatorial properties and lower bounds for knot complexity.
Types of Symmetries of Knotsmath.GT
Classifies symmetries of knots using group actions and orthogonal representation theory.
problem Classifying symmetries of knots in 3-sphere.
method Using geometrization and orthogonal representation theory, constructing examples, and distinguishing symmetries.
result Cyclic and dihedral families of symmetries of prime knots and composite knots.
Hyperbolic Knotsmath.GT
In 1978, W. Thurston revolutionized low diemsional topology with his work on hyperbolic 3-manifolds. In this paper, we discuss what is currently known about knots in the 3-sphere with hyperbolic complements. Then focus is on geometric invariants coming out of the hyperbolic structures. This is one of a collection of ar…
The paper reinterprets knot group invariants using affine transformations.
problem Alexander invariants of knots and their geometric interpretation.
method Representation varieties of knot groups into .
result Alexander polynomial as the singular locus of a coherent sheaf.
Lectures on knot homologymath-ph
We provide various formulations of knot homology that are predicted by string dualities. In addition, we also explain the rich algebraic structure of knot homology which can be understood in terms of geometric representation theory in these formulations. These notes are based on lectures in the workshop "Physics and Ma…
Machine learning maps knots to embeddings, revealing topological invariants.
problem Learning topological invariance in knot theory.
method Contrastive and generative machine learning techniques, auto-regressive decoder Transformer network.
result Neural networks can map different knots to the same point in an embedding vector space.
The paper calculates minimal ribbonlength for various knots.
problem Tabulation of knots based on energy criterion.
method Direct method for computing minimal ribbonlength.
result Computed minimal ribbonlength for specific knots.
Springer varieties appear in both geometric representation theory and knot theory. Motivated by knot theory and categorification Khovanov provides a topological construction of Springer varieties. We extend Khovanov's construction to all two-row Springer varieties. Using the combinatorial and diagrammatic …
We continue our study of the knot Floer homology invariants of cable knots. For large |n|, we prove that many of the filtered subcomplexes in the knot Floer homology filtration associated to the (p,pn+1) cable of a knot, K, are isomorphic to those of K. This result allows us to obtain information about the behavior of …
Lower bounds of betti numbers for homology groups of racks and quandles will be given using the quotient homomorphism to the orbit quandles. Exact sequences relating various types of homology groups are analyzed. Geometric methods of proving non-triviality of cohomology groups are also given, using virtual knots. The r…
A knot K is called n-adjacent to another knot K', if K admits a projection containing n generalized crossings such that changing any 0 < m \leq n of them yields a projection of K'. We apply techniques from the theory of sutured 3-manifolds, Dehn surgery and the theory of geometric structures of 3-manifolds to answer th…
We explain the notion of a grope cobordism between two knots in a 3-manifold. Each grope cobordism has a type that can be described by a rooted unitrivalent tree. By filtering these trees in different ways, we show how the Goussarov-Habiro approach to finite type invariants of knots is closely related to our notion of …
Highly twisted diagramsmath.GT
New proof shows certain knots are hyperbolic.
problem Characterizing knots with highly twisted diagrams.
method New approach involving Euler characteristic argument.
result Complements of certain knots are unannular and atoroidal.
In this paper we discuss the question how matter may emerge from space. For that purpose we consider the smoothness structure of spacetime as underlying structure for a geometrical model of matter. For a large class of compact 4-manifolds, the elliptic surfaces, one is able to apply the knot surgery of Fintushel and St…
The paper connects Apollonian packings to knot theory and improves link representations.
problem Realizing algebraic links in Apollonian packings.
method Introducing new representations of links in tangency graphs of sphere packings, proving link realizability, and improving upper bounds.
result Any algebraic link can be realized in the cubic section of the orthoplicial Apollonian packing.
In this paper, we construct a new homology theory for semi-groups satisfying the self distributivity axiom or the idempotency axiom. Next, we consider the geometric realization corresponding to the homology theory. We continue with the comparison of this homology theory with one term and two term (rack) homology theori…
Computer program finds FAMED triangulations for thousands of knots.
problem Proving the Andersen-Kashaev volume conjecture for many knots.
method Straightforward computer implementation in Regina and Snappy.
result Andersen-Kashaev conjecture proven for over 42,000 knots.
Researchers develop methods to construct Lagrangian cobordisms between Legendrian knots.
problem Understanding the relationship between Legendrian knots through Lagrangian cobordisms.
method Combinatorial and geometric methods, including Heegaard Floer Homology and contact surgery.
result Construction of nondecomposable Lagrangian cobordisms between Legendrian knots.
The paper calculates a formula for knot complements using holomorphic curves.
problem Calculating the partition function of knot complements.
method Skein valued holomorphic curve counting techniques.
result The partition function localizes on specific holomorphic annuli for torus knots.
This paper is a survey on the theory of knotoids and braidoids. Knotoids are open ended knot diagrams in surfaces and braidoids are geometric objects analogous to classical braids, forming a counterpart theory to the theory of knotoids in the plane. We survey through the fundamental notions and existing works on these …
Constructs knots from 3-manifolds with specified geometric limits.
problem Build knots from 3-manifolds with specific geometric properties.
method Constructs explicit families of knots converging to specified 3-manifolds.
result Constructs knots from geometrically finite 3-manifolds with one end.
Develops a framework for stuck knots with rigid constraints and invariants.
problem Rigidity constraints on knot diagrams restrict allowable isotopies.
method Formalizes stuck crossings as rigid configurations, introduces unstick move, and constructs invariants.
result Rigidity contributes independent information even when knot type remains fixed.
Braidoidsmath.GT
Braidoids generalize the classical braids and form a counterpart theory to the theory of planar knotoids, just as the theory of braids does for the theory of knots. In this paper, we introduce basic notions of braidoids, a closure operation for braidoids, we prove an analogue of the Alexander theorem, that is, an algor…
Consider the Chern-Simons topological quantum field theory with gauge group SU(2) and level k. Given a knot in the 3-sphere, this theory associates to the knot exterior an element in a vector space. We call this vector the knot state and study its asymptotic properties when the level is large. The latter vector space b…
Yokota suggested an optimistic limit method of the Kashaev invariants of hyperbolic knots and showed it determines the complex volumes of the knots. His method is very effective and gives almost combinatorial method of calculating the complex volumes. However, to describe the triangulation of the knot complement, he re…