New links homotopic but not isotopic to a given link.
problem Constructing links that are homotopic but not isotopic.
method Constructing a family of links with specific properties.
result Existence of links homotopic but not isotopic to a given link.
Links with isotopic preimages in S3 are isotopic in RP3.
problem Characterizing isotopy in RP3 from S3 preimages. method Analysis of isotopy invariants under double covering map.
result Isotopic preimages in S3 imply isotopic links in RP3. New spheres can split a 4D link in ways not possible in 3D.
problem Exploring how splitting spheres behave in 4D space.
method Constructing specific 2-component surface-links in S4. result Found non-isotopic splitting spheres in S4∖Lm,n. The study finds non-simple isotopy classes of links in 3-manifolds, including Legendrian and pseudo-Legendrian examples.
problem Characterizing isotopy classes of links in 3-manifolds, especially in contact structures.
method Developed theory of links transverse to a nowhere-zero vector field, constructing examples in both Legendrian and pseudo-Legendrian settings.
result Non-simple isotopy classes of links exist, including Legendrian and pseudo-Legendrian examples.
Infinitely many hyperbolic links in lens space have isotopic lifts in 3-sphere.
problem Finding isotopic links in 3-sphere with specific properties.
method Using double covers and Reidemeister moves to construct and analyze links.
result Infinitely many non-isotopic hyperbolic links in lens space have isotopic lifts in 3-sphere.
New link pairs exist that can't be deformed without increasing length.
problem Link isotopy and thick isotopy constraints.
method Constructed isotopic link pairs with preserved total length.
result Link pairs exist that are not thick isotopic.
Basket links are shown to be isotopic to T(2,n+1).
problem Understanding isotopy of basket links to torus links.
method Using symmetrized Seifert form congruence to An matrix. result Basket links are isotopic to T(2,n+1). The paper proves infinitely many non-isotopic splitting 3-spheres for split sphere links in 4D.
problem Proving the existence of infinitely many non-isotopic splitting 3-spheres for split sphere links in 4D.
method Establishing a general sufficient condition for infinitely many topologically non-isotopic splitting 3-spheres in connected sums of 4-manifolds.
result Proves infinitely many non-isotopic splitting 3-spheres for split sphere links in 4D.
Same-genus Seifert surfaces for non-split alternating links are smoothly isotopic.
problem Understanding the uniqueness of Seifert surfaces for non-split alternating links.
method Analyzing the smooth isotopy of Seifert surfaces in the 4-ball.
result Same-genus Seifert surfaces for non-split alternating links are smoothly isotopic.
New link invariants derived from crossing multiplexing.
problem Creating new link invariants from classical to virtual links.
method Multiplexing crossings to create new virtual link diagrams.
result New link invariants derived from known welded link invariants.
Links can be transformed into many others using a specific operation.
problem Understanding the relationship between strongly quasipositive links and their concordance.
method Used a satellite operation with a slice knot to transform links.
result Strongly quasipositive links can be transformed into infinitely many other links.
Study of cable links of uniformly thick knots, revealing new isotopy phenomena.
problem Understanding Legendrian isotopy in cable links of uniformly thick knots.
method Introduced new technique of Legendrian surgeries to classify Legendrian knots in negative cables of twist knots.
result Found new phenomena of stabilized Legendrian links that are smoothly isotopic but not Legendrian isotopic.
New non-isotopic Seifert surfaces found in 4-ball.
problem Finding non-isotopic Seifert surfaces for knots and links.
method Using double branched covers and Seifert forms, with an algorithm based on quadratic forms.
result Families of knots and links with non-isotopic Seifert surfaces in D4. The paper compares isotopic and diffeomorphic links in lens spaces.
problem Comparing isotopic and diffeomorphic links in lens spaces.
method Provides a set of moves on disk, band, and grid diagrams to connect diffeomorphic links.
result There are up to four isotopy equivalent links in each diffeo equivalence class.
New proof of link factorization theorem, avoiding case exhaustion.
problem Proving every non-split link can be uniquely factored into prime links.
method Shorter proof using string links and ambient isotopy.
result Demonstrates existence of string links with no local knots.
A new Markov theorem for 4D ribbon torus-links.
problem Describing isotopic links in R4. method Develops a theorem for ribbon torus-links in B3imesS1. result First step towards a 4D Markov theorem.
Study finds non-isotopic transverse tori in Engel manifolds.
problem Identifying distinct transverse tori in Engel manifolds.
method Constructing an infinite family of non-isotopic transverse tori, introducing a homological invariant.
result Found an infinite family of non-isotopic transverse tori that are smoothly isotopic.
We construct infinite families of topologically isotopic but smoothly distinct knotted spheres in many simply connected 4-manifolds that become smoothly isotopic after stabilizing by connected summing with S2×S2, and as a consequence, analogous families of diffeomorphisms and metrics of positive scalar curva…
Finite type invariants separate PL links in 3D space.
problem If two PL links are isotopic, are they PL isotopic?
method Proved PL isotopic links are indistinguishable by finite type invariants.
result Finite type invariants separate PL links in 3D space if and only if certain conjectures hold.
Theory of link projections to 3-manifold spines, proving combinatorial moves for isotopic links.
problem Link equivalence on 3-manifold spines.
method Theory of link projections, combinatorial moves, Reidemeister Theorem.
result Crossingless projections and combinatorial moves for isotopic links.
New method to classify simple Smale flows on S3.
problem Classifying simple Smale flows on S3. method Embedded template and Kauffman's invariant of spatial graphs.
result Isotopic classification of simple Smale flows on S3. The paper links knot properties in projective space to their fundamental groups.
problem Understanding the fundamental group of knots in projective space.
method Relates knot properties to the fundamental group of the space minus the knot.
result Provides a simple algorithm to find generators and relations for the fundamental group.
Paper generalizes Yamada polynomial to virtual spatial graphs.
problem Generalizing classical knot theory to virtual spatial graphs.
method Topological definition and combinatorial approach for virtual spatial graphs.
result Generalized Yamada polynomial defined and proven invariant.
New measure of knot complexity tied to twist number.
problem Understanding knot complexity.
method Defining alternating volume and showing coarsely equivalent to twist number.
result Alternating volume of a knot is coarsely equivalent to its twist number.
Positive braids have endless filling possibilities.
problem Understanding exact Lagrangian fillings of positive braid links.
method Analyzing positive braid Legendrian links and their fillings.
result Positive braid Legendrian links have infinitely many exact Lagrangian fillings.
The paper studies knitted surfaces and surface-links, showing their isotopy and closure properties.
problem Understanding the isotopy and closure properties of knitted surfaces and surface-links.
method Analyzing the structure and closure of knitted surfaces and surface-links in R4. result Any surface-link is ambient isotopic to the closure of a 2-dimensional knit.
All knots are fused isotopic to the unknot via a process known as virtualization. We extend and adapt this process to show that, up to fused isotopy, classical links are classified by their linking numbers.
We give a classification of n-component links up to Cn-move. In order to prove this classification, we characterize Brunnian links, and have that a Brunnian link is ambient isotopic to a band sum of trivial link and Milnor's links.
Study of strongly invertible Legendrian links in contact 3-space.
problem Characterizing and understanding strongly invertible Legendrian links.
method Equivariant analogs of basic results for strongly invertible and Legendrian links.
result Existence of maximal equivariant Thurston-Bennequin number for strongly invertible links.
Let K be a tunnel number one, fibered link in S^3, with fiber F, and unknotting tunnel τ. We show that τ can be isotoped to lie in F.
Paper constructs infinitely many pairs of Seifert surfaces for each link.
problem Constructing infinitely many pairs of Seifert surfaces for each link.
method Using a multiple group rack (MGR) to construct invariants and distinguishing surfaces using these invariants.
result Presented infinitely many pairs of Seifert surfaces for each link, satisfying specific conditions.
The paper explores isotopic triples of triangles in 3D space.
problem Determining if triples of triangles are combinatorially isotopic.
method Continuous motion of triangles with disjoint outlines, algorithmic checks, and elementary proofs.
result Different types of triples of disjoint triangles are not isotopic.
It is shown that two braids represent transversally isotopic links if and only if one can pass from one braid to another by conjugations in braid groups, positive Markov moves, and their inverses.
New link found that can't be smoothly deformed.
problem Existence of non-isotopic links.
method Constructed a specific link in 3-space.
result Uncountably many I-equivalence classes of links.
Khovanov homology can identify Hopf links.
problem Detecting Hopf links using Khovanov homology.
method Comparing Khovanov homology of links to that of Hopf links.
result Links with matching Khovanov homology are isotopic to Hopf links.
Curves with constant curvature are flexible and can be deformed.
problem Understanding the flexibility of curves with constant curvature.
method Proving the parametric C1-dense relative h-principle for curves of constant curvature. result Two knots of constant curvature are isotopic and homotopic if their self-linking numbers are equal.
Smoothly isotopic 3-discs in 4-sphere become identical after 5-dimensional push.
problem Smooth isotopy of 3-discs in 4-sphere.
method Pushing 3-discs into 5-dimensional space.
result Isotopic 3-discs in 4-sphere become identical after 5-dimensional push.
The abstract discusses knots and their isotopy to the unknot, proving a specific case.
problem Whether every knot is isotopic to the unknot.
method An isotopy extending to a link with specific properties and stronger versions of these results.
result The Bing sling is not isotopic to any PL knot.
We show that a transverse link in a contact structure supported by an open book decomposition can be transversely braided. We also generalize Markov's theorem on when the closures of two braids represent (transversely) isotopic links.
Lisa Traynor has described an example of a two-component Legendrian `circular helix link' in the 1-jet space of the circle (with its canonical contact structure) that is topologically but not Legendrian isotopic to that same link with the order of the two components reversed. We give a complete classification of the Le…
This paper studies isotopies of periodic tangles in 3-manifolds using finite covers.
problem Understanding equivalence relations of periodic tangles in 3-manifolds.
method Employing techniques from 3-manifold topology to study complements of links.
result Isotopies between different finite covers of isotopic links are preserved.
Study equivariant isotopy in higher dimensions, finding exceptions.
problem Under what conditions are equivariant isotopic diffeomorphisms also equivariantly isotopic?
method Construct an invariant valued in the homology of an infinite cover to distinguish isotopic from equivariantly isotopic diffeomorphisms.
result Many equivariant diffeomorphisms are isotopic but not equivariantly isotopic in higher dimensions.
Meridian lemma extended to fully alternating links in thickened surfaces.
problem Extending Menasco's meridian lemma to fully alternating links in thickened surfaces.
method Developed a new meridian lemma for fully alternating links in thickened orientable surfaces of positive genus.
result The meridian lemma holds for fully alternating links in thickened surfaces.
Affine links in projective space have a specific group property.
problem Characterizing links in projective space.
method Examined the fundamental group of link complements.
result Affine links have a non-trivial element of order two in their fundamental group.
New theorem allows transverse links to be braided with rational book structure.
problem Transverse isotopy of links in contact 3-manifolds.
method Generalization of Pavelescu's argument for rational open book decomposition.
result Every transverse link can be isotoped to a braid with rational open book.
An elementary stabilization of a Legendrian link L in the spherical cotangent bundle ST∗M of a surface M is a surgery that results in attaching a handle to M along two discs away from the image in M of the projection of the link L. A virtual Legendrian isotopy is a composition of stabilizations, destabiliz…
Study shows how to create 2-links with any group in 4-manifolds.
problem Creating 2-links with any given group in 4-manifolds.
method Constructing simply connected 4-manifolds containing 2-links with specific properties.
result 2-links can have any group as their 2-link group.
Links on surfaces have at most two intersecting points.
problem Understanding intersections of links on surfaces.
method Analyzing compressing disks and isotopies of surfaces.
result Surfaces can be isotoped to have a standard tube at crossings.