In contrast with knots, whose properties depend only on their extrinsic topology in , there is a rich interplay between the intrinsic structure of a graph and the extrinsic topology of all embeddings of the graph in . For example, it was shown in [2] that every embedding of the complete graph in …
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48 results for “knot embedding”
Knots Inside Fractalsmath.GT
Knots can be embedded into fractals like the Menger Sponge and Sierpinski Tetrahedron.
problem Embedding knots into fractals to compare complexity.
method Proved all knots can be embedded into Menger Sponge, and Pretzel knots into Sierpinski Tetrahedron. Compared iterations needed for given knots.
result Comparison of fractal complexity through knot embedding.
In 1983 Conway and Gordon proved that any embedding of the complete graph into contains at least one nontrivial knot as its Hamiltonian cycle. After their work knots (also links) are considered as intrinsic properties of abstract graphs, and numerous subsequent works have been continued until recen…
This paper characterizes a specific type of twisted Artin groups embedded in knot groups.
problem Embedding twisted right-angled Artin groups in knot groups.
method Defined and characterized twisted right-angled Artin groups through mixed graphs and Klein bottle relations.
result Completely determined which twisted right-angled Artin groups can be embedded in knot groups.
Smoothly knotted 5RP^2 found in 4-sphere.
problem Finding knotted embeddings in higher dimensions.
method Topological and smooth knotting analysis in 4-sphere.
result Smoothly knotted 5RP^2 found in 4-sphere.
The paper characterizes when a 2-sphere can be embedded in a knot trace.
problem Characterizing when a 2-sphere can be embedded in a knot trace.
method Using classical and computable knot invariants, the paper provides conditions for embedding a 2-sphere in a knot trace.
result Conditions for a knot to be topologically -shake slice.
Lickorish has constructed large families of contractible 4--manifolds that have knotted embeddings in the 4--sphere and has also shown that every finitely presented perfect group with balanced presentation occurs as the fundamental group of the complement of a knotted contractible manifold. Here we make a few observati…
To study embeddings of tangles in knots, we use quandle cocycle invariants. Computations are carried out for the tables of knots and tangles, to investigate which tangles may or may not embed in knots in the tables.
We define new invariants of knots by means of quandle colorings and longitudinal information. These invariants can be applied to a tangle embedding problem and recognizing non-classical virtual knots.
Computes knot filtered ECH for torus knots on tight 3-sphere.
problem Computing knot filtered ECH for torus knots.
method Generalized knot filtered ECH definition, Morse-Bott methods, energy filtered Seiberg-Witten theory.
result Computed knot filtered ECH for T(2,q) knots (q odd, positive).
Study how knots occupy space using topological methods.
problem Understanding how knots occupy a volume of space.
method Statistical analysis of persistent homology features from Vietoris-Rips complexes.
result Existence of correlations between geometric and topological features of knots.
The paper generates triangulations of 2-knot complements via spinning 1-knots.
problem Generating triangulations of 2-knot complements.
method Algorithm to generate triangulations of 2-knot complements obtained by spinning 1-knots.
result Triangulations of exteriors of 2-knots from all 1-knots with up to eight crossings.
This paper explores the relationship between the existence of an exact embedded Lagrangian filling for a Legendrian knot in the standard contact $\rr^3$ and the hierarchy of positive, strongly quasi-positive, and quasi-positive knots. On one hand, results of Eliashberg and especially Boileau and Orevkov show that every…
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.
This paper focuses on the graphs in the Petersen family, the set of minor minimal intrinsically linked graphs. We prove there is a relationship between algebraic linking of an embedding and knotting in an embedding. We also present a more explicit relationship for the graph between knotting and linking, whi…
Embedding calculus invariants solve knot connectivity and grope cobordism problems.
problem Knot connectivity and grope cobordism problems.
method Embedding calculus invariants and graph complexes.
result First non-vanishing invariant of grope cobordant knots equals underlying decorated tree equivalence class.
Study of fundamental groups of knotted solenoid complements in 3D sphere.
problem Determining fundamental groups of knotted solenoid complements.
method Using canonical sequence of knot groups and embedding up to mirror reflection.
result Fundamental groups of knotted solenoid complements are solely determined by a sequence of knot groups and embedding up to mirror reflection.
The paper shows conditions under which certain 4-manifolds have no smooth spines.
problem Conditions for 4-manifolds to have no smooth spines.
method Using Heegaard Floer homology and high-dimensional surgery theory, the paper identifies obstructions for 4-manifolds to have smooth spines.
result The paper proves that certain knots and 4-manifolds do not have smooth spines.
New gauge theory invariant detects non-smooth isotopy of -knots.
problem Detecting non-smooth isotopy of -knots.
method Real Seiberg-Witten theory to construct a gauge theoretic invariant.
result Found a family of -knots that are topologically isotopic but not smoothly isotopic.
We give an overview of how calculus of the embedding functor can be used for the study of long knots and summarize various results connecting the calculus approach to the rational homotopy type of spaces of long knots, collapse of the Vassiliev spectral sequence, Hochschild homology of the Poisson operad, finite type k…
We give an infinite family of knots such that for any given , the family contains a knot which can be embedded on a hexagonal -mosaic, but cannot fit on a hexagonal -mosaic in an embedding that achieves its crossing number. This extends the rectangular mosaic result of Ludwig, Evans, and Paat. We also i…
Researchers solve a question about embedding knots into Legendrian structures.
problem Whether every fixed knot type and Legendrian representative have surjective homomorphisms.
method Study of Legendrian embeddings and smooth embeddings in from a homotopical viewpoint.
result Positive answer for infinitely many knot types in three main families, showing rigidity at higher homotopy levels.
Study on knot properties, showing relation between unknotting and crossing numbers.
problem Relations between unknotting and crossing numbers of spatial embeddings.
method Analyzes handcuff-graphs and theta curves, extends known results to handlebody-knots.
result Characterizes handlebody-knots satisfying the equality between unknotting and crossing numbers.
Prime knots with arc index up to 11 and an upper bound of arc index for non-alternating knotsmath.GT
Every knot can be embedded in the union of finitely many half planes with a common boundary line in such a way that the portion of the knot in each half plane is a properly embedded arc. The minimal number of such half planes is called the arc index of the knot. We have identified all prime knots with arc index up to 1…
New invariant for knotted tori, similar to classical invariant.
problem Defining a new topological invariant for knotted tori.
method Analogous to Levine-Tristram invariant, using gauge theory for singular connections.
result Invariant matches Echeverria's invariant and Langte Ma's general result.
Extends exotic embeddings of RP^2 to a larger family and produces homotopy spheres.
problem Constructing exotic embeddings of RP^2 and homotopy spheres.
method Using Montesinos knots and roll-spun knots to prove the existence of homotopy spheres.
result An infinite family of homotopy spheres and homotopy CP^2s are produced.
The study finds infinite knot exteriors with meridional surfaces of any genus and boundary components.
problem Existence of meridional essential surfaces in knot exteriors.
method Analyzing knot exteriors and their surfaces.
result Existence of infinitely many knot exteriors with meridional essential surfaces of any genus and boundary components.
Ribbonness on classical linkmath.GT
Ribbonness proven for slice knots, solving an old question.
problem Proving ribbonness for slice knots.
method Showing a link bounds a ribbon surface that is a renewal embedding of a bounding surface.
result Every slice knot is a ribbon knot.
In this paper, we describe the relation between the study of closed connected surfaces embedded in and the theory of handlebody-knots. By Fox's theorem, a pair of handlebody-knots is associated to a closed connected surface embedded in in the sense that their exterior components are pairwise homeomorphic. W…
On the neighborhood of knotsmath.GT
New framework distinguishes knots via neighborhood invariants.
problem Distinguishing knots and knotoids.
method Study of knotoid spectra and neighborhood invariants.
result Neighborhood invariants can distinguish knots of higher Gordian distance.
Study the spaces of Legendrian knots and links with maximal Thurston-Bennequin invariant.
problem Homotopy types of spaces of Legendrian knots and links with maximal Thurston-Bennequin invariant.
method Recursive formula and contractibility proofs for specific cases.
result Homotopy equivalence and contractibility results for spaces of Legendrian embeddings.
The study finds infinite meridional essential surfaces in hyperbolic knot exteriors of various genera.
problem Existence of meridional essential surfaces in hyperbolic knot exteriors.
method Essential tangle decompositions and meridional essential embeddings.
result Infinite collection of hyperbolic knots with meridional essential surfaces of arbitrary genus and boundary components.
The study determines lens spaces that can be obtained from surgeries on knots in the Poincaré homology sphere.
problem Identifying lens spaces that can be obtained from surgeries on knots in the Poincaré homology sphere.
method Developed a lattice embedding obstruction to realize L-space surgeries on knots in the Poincaré homology sphere.
result Identified the only two knots in the Poincaré homology sphere that admit half-integer lens space surgeries.
3028 obstructions found for embedding without knots.
problem Finding obstructions for knotless embedding.
method Surveying recent work, updating obstructions, and proposing new questions.
result New obstructions with μ=6 and insights into connectivity.
In this paper we introduce a representation of a embedded knotted (sometimes Lagrangian) tori in $\BR^4$ called a hypercube diagram, i.e., a 4-dimensional cube diagram. We prove the existence of hypercube homology that is invariant under 4-dimensional cube diagram moves, a homology that is based on knot Floer homology.…
We construct knotted proper holomorphic embeddings of the unit disc in C^2.
The paper extends knot theory to 4-manifolds, defining new genera and obstructions.
problem Understanding embeddings of 3-manifolds into 4-manifolds with specific properties.
method New -signature obstructions and extensions of knot genera.
result Existence of knots with arbitrarily large generalized superslice genera and double stabilizing numbers.
Study transverse knots and symplectic surfaces using Seiberg-Witten monopole equations.
problem Investigate transverse knots and symplectic surfaces via Seiberg-Witten monopole equations.
method Equivariant Seiberg-Witten theory on branched covers, introducing a novel slice-torus invariant.
result Determine the value of slice-torus invariant for certain Montesinos knots within a deviation of 2.
A Weiss-Williams theorem for spaces of embeddings and the homotopy type of spaces of long knotsmath.AT
The paper establishes a new pseudoisotopy result for embedding spaces, leading to computations of homotopy groups of long knots.
problem Computing homotopy groups of spaces of long knots in high codimension.
method Using pseudoisotopy results and algebraic K-theory, the paper describes the difference in homotopy types of block and ordinary embeddings of a codimension at least three embedding.
result The homotopy type of spaces of long knots of codimension at least 3 is determined explicitly, including torsion information.
For a positive integer , the collection of -sided polygons embedded in -space defines the space of geometric knots. We will consider the subspace of equilateral knots, consisting of embedded -sided polygons with unit length edges. Paths in this space determine isotopies of polygons, so path-components …
Smooth knots can be embedded into a specific Menger continuum.
problem Embedding smooth knots into a specific type of continuum.
method Explicit construction using cubical models and self-similarity of the Menger continuum.
result Every smooth knot can be isotoped into the Menger continuum.
We construct a graph G such that any embedding of G into R^{3} contains a nonsplit link of two components, where at least one of the components is a nontrivial knot. Further, for any m < n we produce a graph H so that every embedding of H contains a nonsplit n component link, where at least m of the components are nont…
Study of 3-manifolds in 5-sphere using bridge decompositions.
problem Understanding embeddings of 3-manifolds in 5-sphere.
method Introduce and study bridge decompositions, use multisections of 5-manifolds.
result Every embedded 3-manifold admits a bridge decomposition.
In "Width complexes for knots and 3-manifolds," Jennifer Schultens defines the width complex for a knot in order to understand the different positions a knot can occupy in the 3-sphere and the isotopies between these positions. She poses several questions about these width complexes; in particular, she asks whether the…
Fibered ribbon pretzelsmath.GT
Classifies fibered ribbon pretzels, except for a few cases.
problem Classifying fibered ribbon pretzel knots up to mutation.
method Combining lattice embedding techniques with Gabai's classification of fibered pretzel knots, and exhibiting ribbon disks.
result Complete classification except for a few cases.
We investigate properties of spatial graphs on the standard torus. It is known that nontrivial embeddings of planar graphs in the torus contain a nontrivial knot or a nonsplit link due to [1],[2]. Building on this and using the chirality of torus knots and links [3],[4], we prove that nontrivial embeddings of simple 3-…
A knot k in a closed orientable 3-manifold is called nonsimple if the exterior of k possesses a properly embedded essential surface of nonnegative Euler characteristic. We show that if k is a nonsimple prime tunnel number one knot in a lens space M (where M does not contain any embedded Klein bottles), then k is a (1,1…
New research finds six bipartite intrinsically knotted graphs with 23 edges.
problem Identifying intrinsically knotted bipartite graphs with 23 edges.
method Analyzing embeddings and graph minors to find minimal intrinsically knotted graphs.
result No minor minimal intrinsically knotted bipartite graph exists with 23 edges.