Study of knots and links in 2-complexes, defining linking numbers and polynomials.
problem Understanding knots and links in 2-dimensional complexes.
method Definition of linking numbers and Kauffman-type bracket polynomials for links in 2-complexes.
result Established relationships between 2-complexes and knots/links in 3-manifolds.
We define and prove properties of link lattice complexes for plumbed links.
problem Understanding the homology and formality of plumbed L-space links.
method Define and analyze link lattice complexes, proving homotopy equivalence and formality properties.
result Link lattice complexes are homotopy equivalent to link Floer complexes for plumbed links.
The paper studies the minimal number of clasps in C-complexes for links.
problem Determining the minimal number of clasps in C-complexes for links.
method Examining C-complexes bounded by links, focusing on the number of clasps and their relation to link invariants.
result For 2-component links with nonzero linking number, the linking number determines the minimal number of clasps.
An additional minimal simplicial n-complex contains a non-splittable link in R^(2n).
problem Constructing minimal simplicial n-complexes with a specific link property.
method Presenting a new minimal simplicial n-complex.
result An additional simplicial n-complex with the same link property.
The primary objects of study in the ``knot theory of complex plane curves'' are C-links: links (or knots) cut out of a 3-sphere in the complex plane by complex plane transverse and totally tangential. Transverse C-links are naturally oriented. There are many natural classes of examples: links of singularities; links at…
Kervaire's sphere-link is equivalent to a ribbon sphere-link, simplifying complex 2-complexes.
problem Understanding the structure of 2-complexes and their asphericity.
method Using Kervaire's sphere-link and ribbon sphere-link equivalence, analyzing the compact complement of ribbon disk-links.
result Every connected subcomplex of a contractible finite 2-complex is aspherical.
New bounds on clasp number for 3-component links.
problem Measuring how close a 3-component link is to being a boundary link.
method Using C-complexes and minimizing the area of word curves.
result Constructs links achieving new lower bounds for clasp number.
New infinite family of 2-complexes intrinsically linked in 4D.
problem Intrinsic linking of 2-complexes in 4D.
method Examining suspensions of graphs containing K6 as a minor.
result Embeddings of suspensions contain intrinsically linked cycles.
Any two knots admit orientation preserving homeomorphic Seifert surfaces, as can be seen by stabilizing. There is a generalization of a Seifert surface to the setting of links called a C-complex. In this paper, we ask when two links will admit orientation preserving homeomorphic C-complexes. In the case of 2-component …
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.
New transformations preserve link isotopy, showing complexity differences.
problem Link isotopy preservation with complexity differences.
method Introducing multiflypes of rectangular diagrams of links.
result Two diagrams of same complexity not related by simpler moves.
Flexible links have symplectic representatives in complex projective space.
problem Existence of symplectic representatives for flexible links.
method Construction of a symplectic surface invariant under complex conjugation.
result No obstructions to finding symplectic representatives beyond classical topology.
Minimal simplicial complexes in high dimensions always contain complex links.
problem Existence of complex links in high-dimensional embeddings.
method Demonstrated through minimal simplicial complexes in R2n. result Minimal simplicial n-complexes inevitably contain a nonsplittable two-component link. Link condition for simplicial complexes to be CUB spaces.
problem Understanding when a simplicial complex is a CUB space.
method Establishing a link condition based on local lattice properties.
result The link condition generalizes Gromov's link condition for cube complexes.
We explore the complex associated to a link in the geometric formalism of Khovanov's (n=2) link homology theory, determine its exact underlying algebraic structure and find its precise universality properties for link homology functors. We present new methods of extracting all known link homology theories directly from…
Improved lower bounds on 2-bridge link complexity.
problem Finding minimal triangulations of 2-bridge link complements.
method Explicit angle structures and volume estimates.
result Explicit lower bounds on link complexity.
This paper computes Kakimizu complexes for all 11 crossing prime alternating knots.
problem Computing Kakimizu complexes for prime, alternating knots with 11 crossings.
method Used known algorithms and Murasugi sums with sutured manifold theory.
result Explicitly described Kakimizu complexes for all 11 crossing prime alternating knots.
We give a characterization of alternating link exteriors in terms of cubed complexes. To this end, we introduce the concept of a "signed BW cubed-complex", and give a characterization for a signed BW cubed-complex to have the underlying space which is homeomorphic to an alternating link exterior.
We determine the algebraic structure underlying the geometric complex associated to a link in Bar-Natan's geometric formalism of Khovanov's link homology theory (n=2). We find an isomorphism of complexes which reduces the complex to one in a simpler category. This reduction enables us to specify exactly the amount of i…
Researchers use estimated Kolmogorov complexity for better link prediction in graphs.
problem Improving link prediction accuracy in complex networks.
method Regularization based on an approximation of Kolmogorov complexity, which is differentiable and compatible with recent link prediction algorithms.
result The regularization method shows good performance on diverse real-world networks, but the success is likely due to an aggregation method rather than actual estimation of Kolmogorov complexity.
Study on entanglement complexity of confined ring polymers in lattice tubes.
problem Understanding the entanglement complexity of confined ring polymers in lattice tubes.
method Applied knot theory to extend and prove results about the complexity of 2SAPs.
result Proved that all but exponentially few size m 2SAPs have F complexity that grows at least linearly in m as m approaches infinity.
We consider closed simplicial and cubical n-complexes in terms of link of their (n−2)-faces. Especially, we consider the case, when this link has size 3 or 4, i.e., every (n−2)-face is contained in 3 or 4 n-faces. Such simplicial complexes with {\em short} (i.e. of length 3 or 4) links are completely classified…
Fix an integer N>1. To each diagram of a link colored by 1,...,N, we associate a chain complex of graded matrix factorizations. We prove that the homotopy type of this chain complex is invariant under Reidemeister moves. When every component of the link is colored by 1, this chain complex is isomorphic to the chain com…
We analyze the effect of adding, removing, and moving basepoints on link Floer homology. We prove that adding or removing basepoints via a procedure called quasi-stabilization is a natural operation on a certain version of link Floer homology, which we call CFLUV∞. We consider the effect on the full link Flo…
Define the complete n-complex on N vertices to be the n-skeleton of an (N-1)-simplex. We show that embeddings of sufficiently large complete n-complexes in R^{2n+1} necessarily exhibit complicated linking behaviour, thereby extending known results on embeddings of large complete graphs in R^3 (the case n=1) to higher d…
We present the complex analytic and principal complex analytic realizability of a link in a 3-manifold M as a tool for understanding the complex structures on the cone C(M).
Paper confirms Whitehead's conjecture for aspherical 2-complexes.
problem Whitehead's conjecture about aspherical 2-complexes.
method Argument on ribbon sphere-links, generalized for aspherical 2-complexes.
result Whitehead's conjecture confirmed for aspherical 2-complexes.
Formula for satellite operators using knot Floer homology.
problem Computing knot Floer homology for satellite knots.
method Using Heegaard Floer Dehn surgery formulas and formal knot Floer complexes.
result Formulas to compute knot Floer homology for satellite knots.
New simplicial complexes show unavoidable link of spheres in high dimensions.
problem Finding unavoidable links of spheres in high-dimensional spaces.
method Simple argument in piecewise linear topology and application of the van Kampen--Flores theorem.
result Existence of additional simplicial complexes with unavoidable links of spheres.
To a presentation of an oriented link as the closure of a braid we assign a complex of bigraded vector spaces. The Euler characteristic of this complex (and of its triply-graded cohomology groups) is the HOMFLYPT polynomial of the link. We show that the dimension of each cohomology group is a link invariant.
We give upper bounds of the Matveev complexities of two-bridge link complements by constructing their spines explicitly. In particular, we determine the complexities for an infinite sequence of two-bridge links corresponding to the continued fractions of the form [2,1,...,1,2]. We also give upper bounds for the 3-manif…
Link Floer homology is split into snake complexes and local systems.
problem Classifying link Floer complexes over specific rings.
method Classifying isomorphism and chain homotopy equivalence classes of free chain complexes over a specific ring, then applying these results to link Floer complexes.
result Link Floer complexes split uniquely into snake complexes and local systems.
The abstract describes a strategy to construct reduced Khovanov homology for links in lens spaces.
problem Constructing reduced Khovanov homology for links in lens spaces.
method Generalizing a symplectic interpretation of reduced Khovanov homology for links in S3 and constructing cochain complexes for links in S3 and S2imesS1. result The cohomology of the constructed cochain complex for links in S2imesS1 may be a link invariant. New linking numbers link complex cycles to Calabi-Yau 3-folds.
problem Understanding complex analytic cycles and their Massey products.
method Relating ABC Massey products to holomorphic linking numbers.
result Constructed a family of Calabi-Yau 3-folds with non-vanishing Massey products.
Corrected a false lemma in Cimasoni's work on linking theory.
problem A false lemma in Cimasoni's geometric construction of the Conway potential function.
method Presented counterexamples and a detailed proof of the corrected lemma.
result The lemma is false and its correction has significant consequences for subsequent works.
New link groups are derived from torus necklaces, connecting braid groups to reflection groups.
problem Understanding the relationship between braid groups and reflection groups.
method Constructing torus necklaces and linking them to braid groups of J-reflection groups. result Link groups of torus necklaces are precisely braid groups of J-reflection groups, with meridians as braid reflections. Visual construction of maps linking to two-bridge links.
problem Understanding stable maps and their properties for two-bridge links.
method Visual construction and analysis of stable maps from 3-sphere to plane.
result Determination of stable map complexities for some two-bridge link exteriors.
New links found that challenge traditional complexity assumptions.
problem Traditional complexity assumptions in concordance of links are challenged.
method Study of cobordisms between links in thickened surfaces, use of augmented Khovanov homology.
result Existence of concordant links that require thickenings of higher genus surfaces.
Circle graph complexes reveal link properties via Khovanov homology.
problem Understanding topological properties of links using circle graph complexes.
method Analyzing homotopy types and computing Khovanov homology of specific knots.
result Real extreme Khovanov homology of a 4-strand pretzel knot computed.
There are two natural simplicial complexes associated to the noncrossing partition lattice: the order complex of the full lattice and the order complex of the lattice with its bounding elements removed. The latter is a complex that we call the noncrossing partition link because it is the link of an edge in the former. …
Study annular concordance invariants and refine transverse link invariants.
problem Understanding annular concordance and transverse links.
method Defined and studied annular concordance invariant, introduced modified chain complex, and refined transverse invariant.
result Refined transverse invariant and studied its relationship with transverse and braid monodromy properties.
Polynomial bound on Reidemeister moves for each link type.
problem Recognizing whether a given link diagram represents a specific link type.
method Showed existence of a polynomial pK such that any two diagrams of a link type differ by at most pK(c1)+pK(c2) Reidemeister moves. result The problem of recognising a link type is in NP and can be completed in exponential time.
The slope invariant is shown to be unchanged by concordance of colored links.
problem Describing the concordance of colored links with a distinguished component.
method Proved the slope invariant is invariant under concordance using C-complexes and generalized Seifert forms.
result The slope invariant is invariant under concordance of links.
We develop an algorithm of polynomial time complexity to construct the Grushko decomposition of fundamental groups of graphs of free groups with cyclic edge groups. Our methods rely on analysing vertex links of certain CAT(0) square complexes naturally associated with a special class of the above groups. Our main resul…
Study differentiable maps on hypersurface links, finding fold maps with circle singular value sets.
problem Understanding differentiable maps on hypersurface links.
method Restricting holomorphic functions to hypersurface links and analyzing the resulting maps.
result Found fold maps with concentric circle singular value sets.
ConEx learns complex embeddings for knowledge graphs, improving link prediction.
problem Predicting missing links in knowledge graphs.
method 2D convolution with Hermitian inner product of complex-valued embeddings.
result ConEx outperforms state-of-the-art methods on various benchmarks.
New formulas link Milnor invariants to Heegaard Floer homology.
problem Understanding Milnor invariants through Heegaard Floer homology.
method Established new relationships between Milnor invariants and Heegaard Floer homology.
result Formula for the Milnor triple linking number from the link Floer complex.
For knots in S^3, the bi-graded hat version of knot Floer homology is defined over Z; however, for a link L in S^3 with #|L|=l>1, there are 2^{l-1} bi-graded hat versions of link Floer homology defined over Z, the multi-graded hat version of link Floer homology is only defined over F_2 from holomorphic considerations, …