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.
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. New links split by integer homology spheres but not by others.
problem Characterizing links split by integer homology spheres.
method Constructing specific links and homology spheres.
result Infinite families of links and homology spheres split by specific ones but not by others.
Researchers create infinite Brunnian links of 3-balls in 4-sphere.
problem Constructing infinite Brunnian links of 3-balls in 4-sphere.
method Using the third author's result on splitting spheres for trivial 2-spheres link in 4-sphere, and providing a new proof.
result Infinitely many n-component Brunnian links of 3-balls in 4-sphere constructed.
We prove properties of linking forms on rational homology spheres.
problem Properties of linking forms on rational homology spheres.
method Use of Heegaard splittings and properties of Q/Z-valued linking forms. result Linking forms on rational homology spheres are symmetric or anti-symmetric.
Study Goeritz groups of link decompositions, focusing on their asymptotic behavior.
problem Understanding the asymptotic behavior of Goeritz groups for link decompositions.
method Defined Goeritz groups for link decompositions, analyzed their properties, and discussed their asymptotic behavior.
result Discussed the asymptotic behavior of minimal pseudo-Anosov entropies and related it to Goeritz groups of Heegaard splittings.
The paper studies invariants of surfaces in the 3-sphere using handlebody-links.
problem Understanding invariants of surfaces in the 3-sphere.
method Using Heegaard splittings and G-families of quandles to construct invariants. result Invariants can distinguish certain surfaces in the 3-sphere.
In this paper, we give an isotopy classification of 3-bridge spheres of 3-bridge arborescent links, which are not Montesinos links. To this end, we prove a certain refinement of a theorem of J.S. Birman and H.M. Hilden on the relation between bridge presentations of links and Heegaard splittings of 3-manifolds. In the …
For any link of two components in an integral homology sphere, we define an instanton Floer homology whose Euler characteristic is the linking number between the components of the link. We relate this Floer homology to the Kronheimer-Mrowka instanton Floer homology of knots. We also show that, for two-component links i…
We show that every p-fold strictly-cyclic branched covering of a b-bridge link in the 3-sphere admits a p-symmetric Heegaard splitting of genus g=(b-1)(p-1). This gives a complete converse to a result of Birman and Hilden, and gives an intrinsic characterization of p-symmetric Heegaard splittings as p-fold strictly-cyc…
Identifies a mod-p triple cup product for rational homology 3-spheres with specific first homology.
problem Locally flat embeddings in S4 for rational homology 3-spheres method Using triple torsion linking form and torsion-linking duality
result Identifies the mod-p triple cup product for specific rational homology 3-spheres Explains historical connections between vector bundle splitting and Riemann-Hilbert problems.
problem Vector bundle splitting over the Riemann sphere.
method Historical overview and connections to other mathematical problems.
result Explains the Riemann-Hilbert-Birkhoff problems and their relation to vector bundle splitting.
We prove that any knot or link in any 3-manifold can be nicely decomposed (splitted) by a filling Dehn sphere. This has interesting consequences in the study of branched coverings over knots and links. We give an algorithm for computing Johansson diagrams of filling Dehn surfaces out from coverings of 3-manifolds branc…
Boring is an operation which converts a knot or two-component link in a 3--manifold into another knot or two-component link. It generalizes rational tangle replacement and can be described as a type of 2--handle attachment. Sutured manifold theory is used to study the existence of essential spheres and planar surfaces …
It is shown that if the exterior of a link L in the three sphere admits a genus 2 Heegaard splitting, then L has Generalized Property R.
Extends Seifert algorithm to 3-manifolds via surgery.
problem Construct Seifert surfaces in arbitrary 3-manifolds.
method Uses surgery on framed links in S^3 to extend classical algorithm.
result Explicit construction of Seifert surfaces in 3-manifolds.
Standard position for surfaces extended to weakly generalized alternating links.
problem Proving standard position for surfaces in a broader class of links.
method Extending techniques from classical alternating links to weakly generalized alternating links.
result All weakly generalized alternating links are prime, and Conway spheres interact with diagrams as in classical setting.
A closed, orientable, splitting surface in an oriented 3-manifold is a topologically minimal surface of index n if its associated disk complex is (n−2)-connected but not (n−1)-connected. A critical surface is a topologically minimal surface of index 2. In this paper, we use an equivalent combinatorial definit…
Banchoff's sphere splits trefoil knots in 3-manifolds.
problem Understanding how Dehn spheres split trefoil knots in 3-manifolds.
method Constructing and analyzing filling Dehn spheres and their diagrams.
result Banchoff's sphere splits the trefoil knot in S3. Generalizes Seifert algorithm to integral homology spheres.
problem Finding Seifert circles for integral homology spheres.
method Planar diagram representation of Heegaard splitting, tangle projection.
result Natural construction of Seifert circles and bands for integral homology spheres.
We first present three graphic surgery formulae for the degree n part Zn of the Kontsevich-Kuperberg-Thurston universal finite type invariant of rational homology spheres. Each of these three formulae determines an alternate sum of the form ∑I⊂N(−1)♯IZn(MI) where N is the set of com…
Study homotopy groups of spaces of long links and knots, finding new generators.
problem Understanding homotopy groups of spaces of long links and knots.
method Graphing map increases dimensions, split injections from homotopy groups of spheres, and analyzing knotting effects.
result Generators for homotopy groups in a new degree for spaces of equidimensional long links.
Study spaces of knots and links in specific 3-manifolds.
problem Determine the homotopy types of spaces of knots and links in various 3-manifolds.
method Recursive determination of homotopy types, using fundamental groups and quotient spaces.
result Homotopy types of spaces of knots in solid torus and thickened torus are determined.
We show that if a split link is obtained from a split link L in S3 by 1/n-Dehn surgery along a trivial knot C, then the link L∪C is splittable. That is to say, it is impossible to obtain a split link from a split link via a non-trivial twisting. As its corollary, we completely determine when a trivial li…
Infinitely many splitting spheres found for unlinked 2-spheres in 4-space.
problem Existence of pairwise non-isotopic splitting spheres for unlinked 2-spheres in 4-space.
method Analytical proof showing non-isotopic spheres.
result Infinitely many non-isotopic splitting spheres found.
New methods convert complex link presentations to simpler, recognizable forms.
problem Representing split and composite links in a clear, recognizable way.
method Pocket and flip moves to transform link presentations.
result Pocket moves are the only obstruction to representing split links by split plat presentations.
The study establishes a link between the complexity of fibered knots and the genus of their Heegaard splittings.
problem Understanding the complexity of Heegaard splittings induced by fibered knots.
method Analyzing the monodromy of fibered knots and their impact on Heegaard splittings.
result Minimal genus Heegaard splittings of a three-manifold are unique and can be induced by fibered knots with complex monodromies.
Link Floer homology detects split links.
problem Detecting split links in link Floer homology.
method Module structure on link Floer homology and sutured Floer homology results.
result Link Floer homology detects split links.
The splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new techniques to compute the splitting number, involving covering links and Alexander invariants. As an application, we completely determine the sp…
Algorithm constructs reducing spheres for genus-2 Heegaard splitting of S^3.
problem Finite generation of Goeritz group G2. method Algorithm to construct reducing spheres from a standard reducing sphere.
result Alternate proof of finite generation of G2. Prime homology detects split links in prime characteristic.
problem Detecting split links in prime characteristic.
method Uses Dowlin's spectral sequence and sutured Floer homology with twisted coefficients.
result Reduced sl(P) link homology detects split links in Z/P. New methods for delta-moves on algebraically split links identified.
problem Understanding delta-moves on algebraically split links.
method Introducing self and mixed delta-moves, proving equivalence, and calculating delta-splitting numbers.
result Two links are mixed delta-equivalent if they have the same pairwise linking number and components.
In this paper, we define a lassoing on a link, a local addition of a trivial knot to a link. Let K be an s-component link with the Conway polynomial non-zero. Let L be a link which is obtained from K by r-iterated lassoings. The complete splitting number split(L) is greater than or equal to r+s-1, and less than or equa…
The splitting number of a link is the minimum number of crossing changes between distinct components that is required to convert the link into a split link. We provide a bound on the splitting number in terms of the four-genus of related knots.
The paper extends keenness concept to bridge splittings and finds conditions for existence.
problem Extending keenness concept to bridge splittings and finding conditions for existence.
method Extending the concept of keenness to bridge splittings and proving existence conditions.
result Existence of strongly keen (g,b)-splitting of a link with distance n for certain integers g, b, and n. 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.
Confirming the Powell Conjecture for genus-3 Heegaard splittings of the 3-sphere.
problem Proving the finitely generated nature of the Goeritz group for genus-3 Heegaard splittings of the 3-sphere.
method Establishing the connectivity of reducing sphere complexes for the genus-3 case.
result Confirmation of the Powell Conjecture for genus-3 Heegaard splittings of the 3-sphere.
We study the Kakimizu complex of a split link. As part of this, we also study Seifert surfaces and the Kakimizu complex for a non-split link in a 3-ball. In addition, we show that a simplex of the Kakimizu complex of a non-split link can be realised in an essentially unique way.
The paper studies splitting maps in link Floer homology using skein exact sequences.
problem Understanding splitting maps for links using link Floer homology.
method Link Floer homology and skein exact sequences.
result Splitting maps for torus links T(n,n) are associated with integer points in permutahedra. Study shows how Khovanov homology behaves for split links and cobordisms.
problem Understanding Khovanov homology's sensitivity to cobordisms between split links.
method Proved that Khovanov homology map is determined by individual components of cobordism, not linking.
result Khovanov homology detects non-split links from split links.
From a fibered link in the 3-sphere may be constructed a field of not everywhere tangent 2-planes; when the fibered link is the link of an isolated critical point of a map from 4-space to the plane, the plane field is essentially the field of kernels of the derivative of the map. Homotopically, such a plane field deter…
Classifies involutions on alternating prime non-split links.
problem Classifying involutions on alternating prime non-split links.
method Using the category of flypes between reduced alternating diagrams.
result Quotient of an alternating periodic link is alternating; freely 2-periodic alternating links have an even number of components.
Disk surgery on primitive disks of genus-3 Heegaard splittings of 3-sphere yields no primitive disks.
problem Characterizing primitive disks in genus-3 Heegaard splittings of 3-sphere.
method Analyzing the effect of disk surgery on primitive disks in genus-3 Heegaard splittings of 3-sphere.
result Primitive disks in genus-3 Heegaard splittings of 3-sphere are not weakly closed under disk surgery.
Diagrammatic method characterizes non-split surfaces in 3-sphere.
problem Characterizing non-split compact surfaces in the 3-sphere.
method Using diagrams of spatial trivalent graphs with signs and Reidemeister moves.
result Two diagrams of embedded surfaces are related by Reidemeister moves if and only if the surfaces are ambient isotopic.
The paper establishes conditions for link invariants to bound the weak splitting number.
problem Determining the minimum number of crossing changes to split a link into knots.
method Conditions on R-valued link invariants, link Floer homology, and known obstructions. result New bounds on the weak splitting number for many prime links.
Study shows that splitting links requires an arbitrarily large number of extra crossings.
problem The problem is to determine the minimum number of extra crossings needed to transform a diagram of a split link into a split diagram.
method The approach uses Reidemeister moves and the framework of bubble tangles, along with techniques from Riemannian geometry.
result There exist split links with diagrams requiring an arbitrarily large number of extra crossings.
The paper examines when 2-string tangles can be embedded into specific link types.
problem When 2-string tangles can be embedded into the unknot, unlink, or split links.
method Geometric characterizations, tangle sums, and colorings.
result Prime 2-string tangles with up to seven crossings are classified for embedding into specific link types.
Transformations between different analytic descriptions of constant mean curvature (CMC) surfaces are established. In particular, it is demonstrated that the system \[ \begin{split} &\partial ψ_{1} = (|ψ_{1}|^{2} + |ψ_{2}|^{2}) ψ_{2} \\ &\bar{\partial} ψ_{2} =- (|ψ_{1}|^{2} + |ψ_{2}|^{2}) ψ_{1} \end{split} \] descripti…