arXiv research

This paper generalizes octahedral decomposition to links in thickened surfaces.

Researchers compute Goeritz groups for all (1,1)-link decompositions.

Finite Goeritz groups for links with long bridge decompositions.

Decomposes string links in a surface into prime components.

Geometrically describes hyperbolic structures on link complements using quantum groups.

In this paper we use 3-manifold techniques to illuminate the structure of the string link monoid. In particular, we give a prime decomposition theorem for string links on two components as well as give necessary conditions for string links to commute under the stacking operation.

We construct a Seifert surface for a given null-homologous transverse link in a contact manifold that is compatible with a planar open book decomposition, then obtain a formula of the self-linking number. It extends Bennequin's self-linking number formula for braids in the standard contact 3-sphere.

We find a self-linking number formula for a given null-homologous transverse link in a contact manifold that is compatible with either an annulus or a pair of pants open book decomposition. It extends Bennequin's self-linking formula for a braid in the standard contact $3$-sphere.

This paper classifies fibered links in 3-sphere using open book decompositions.

This work classifies belted sum decompositions of fully augmented links.

We define a decomposition of link projections whose pieces we call atoroidal graphs. We describe a surgery operation on these graphs and show that all atoroidal graphs can be generated by performing surgery repeatedly on a family of well known link projections. This gives a method of enumerating atoroidal graphs and he…

Hyperbolic links in thickened torus decompose into angled tetrahedra.

The A-B slice problem is a reformulation of the topological 4-dimensional surgery conjecture in terms of decompositions of the 4-ball and link homotopy. We show that link groups, a recently developed invariant of 4-manifolds, provide an obstruction for the class of model decompositions, introduced by M. Freedman and X.…

Study Goeritz groups of link decompositions, focusing on their asymptotic behavior.

Study the JSJ-decomposition of a specific 3-manifold.

We consider knots whose diagrams have a high amount of twisting of multiple strands. By encircling twists on multiple strands with unknotted curves, we obtain a link called a generalized augmented link. Dehn filling this link gives the original knot. We classify those generalized augmented links that are Seifert fibere…

New theorem allows transverse links to be braided with rational book structure.

New method characterizes thin links via Conway spheres and tangle decompositions.

We give an upper bound on the z-degree of the Kauffman polynomial of a link, using bridges of length greater than one which are separated in some tangle decomposition of a link diagram. We construct some examples by wiring together rational tangles.

Tied links and the tied braid monoid were introduced recently by the authors and used to define new invariants for classical links. Here, we give a version purely algebraic-combinatoric of tied links. With this new version we prove that the tied braid monoid has a decomposition like a semi--direct group product. By usi…

New invariants derived from link homology for 4-manifolds.

Matrix factorizations and their extensions to tensor factorizations and decompositions have become prominent techniques for linear and multilinear blind source separation (BSS), especially multiway Independent Component Analysis (ICA), NonnegativeMatrix and Tensor Factorization (NMF/NTF), Smooth Component Analysis (Smo…

Decomposing knots and links into tangles is a useful technique for understanding their properties. The notion of prime tangles was introduced by Kirby and Lickorish in [3]; Lickorish proved [5] that by summing prime tangles one obtains a prime link. In a similar spirit, summing two prime alternating tangles will produc…

Planar decomposition simplifies HOMFLY polynomial calculation for certain knots and links.

This paper continues the study of decompositions of a smooth 4-manifold into two handlebodies with handles of index $\leq2$. Part I gave existence results in terms of spines and chain complexes over the fundamental group of the ambient manifold. Here we assume that one side of a decomposition has larger fundamental gro…

Generalizing previous constructions, we present a dual pair of decompositions of the complement of a link L into bipyramids, given any multi-crossing projection of L. When L is hyperbolic, this gives new upper bounds on the volume of L given its multi-crossing projection. These bounds are realized by three closely rela…

We extend the Kamada-Miyazawa polynomial to virtual singular links, which is valued in $\mathbb{Z}[A^2, A^{-2}, h]$. The decomposition of the resulting polynomial into two components, one in $\mathbb{Z}[A^2, A^{-2}]$ and the other in $\mathbb{Z}[A^2, A^{-2}]h$ yields the decomposition of the Kauffman-Jones polynomial o…

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…

Article generalizes open book construction for 5D contact pairs.

It is known by A. Loi and R. Piergallini that a closed, oriented, smooth 3-manifold is Stein fillable if and only if it has a positive open book decomposition. In the present paper we will show that for every link L in a Stein fillable 3-manifold there exists an additional knot L' to L such that the union of the links …

We show that the fundamental quandle defines a functor from the oriented tangle category to a suitably defined quandle category. Given a tangle decomposition of a link $L$, the fundamental quandle of $L$ may be obtained from the fundamental quandles of tangles. We apply this result to derive a presentation of the funda…

For n-bridge decompositions of links in S^3, we propose a practical method to ensure that the Hempel distance is at least two.

New examples of Calabi-Yau metrics on cones with irregular smooth links.

We propose the Relational Tucker3 (RT) decomposition for multi-relational link prediction in knowledge graphs. We show that many existing knowledge graph embedding models are special cases of the RT decomposition with certain predefined sparsity patterns in its components. In contrast to these prior models, RT decouple…

This paper is a survey of some of the most elementary consequences of the JSJ-decomposition and geometrization for knot and link complements in the 3-sphere. Formulated in the language of graphs, the result is the construction of a bijective correspondence between the isotopy classes of links in $S^3$ and a class of ve…

The paper shows knots can have zero support genus in certain spaces.

The study finds counterexamples to a conjecture about incompressible planar surfaces in hyperbolic link exteriors.

We introduce a generalization of the Lisca-Ozsváth-Stipsicz-Szabó Legendrian invariant $\mathfrak L$ to links in every rational homology sphere, using the collapsed version of link Floer homology. We represent a Legendrian link $L$ in a contact 3-manifold $(M,ξ)$ with a diagram $D$, given by an open book decomposition …

The hyperbolic volume of a link complement is known to be unchanged when a half-twist is added to a link diagram, and a suitable 3-punctured sphere is present in the complement. We generalize this to the simplicial volume of link complements by analyzing the corresponding toroidal decompositions. We then use it to prov…

In this paper we write explicitly the open book decompositions of links of quotient surface singularities supporting the corresponding unique Milnor fillable contact structure. The page-genus of these Milnor open books are minimal among all Milnor open books supporting the same contact structure. We also investigate wh…

In this paper, we study the generalized volume conjecture for the colored Jones polynomials of links with complements containing more than one hyperbolic piece. First of all, we construct an infinite family of prime links by considering the cabling on the figure eight knot by the Whitehead chains. The complement of the…

We bound the hyperbolic volumes of a large class of knots and links, called homogeneously adequate knots and links, in terms of their diagrams. To do so, we use the decomposition of these links into ideal polyhedra, developed by Futer, Kalfagianni, and Purcell. We identify essential product disks in these polyhedra.

We solve the ANOVA decomposition for categorical inputs.

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.

A new tensor-based method for predicting temporal relationships in knowledge bases.

We present explicit geometric decompositions of the complement of tiling links, which are alternating links whose projection graphs are uniform tilings of the 2-sphere, the Euclidean plane or the hyperbolic plane. This requires generalizing the angle structures program of Casson and Rivin for triangulations with a mixt…

This paper solves the structure of link concordance groups, proving they are infinitely generated.

Two subset germs of Euclidean spaces are called blow-spherically equivalent, if their spherical modifications are homeomorphic and the homeomorphism induces homeomorphic tangent links. Blow-spherical equivalence is stronger than the topological equivalence but weaker than the Lipschitz equivalence. We introduce the thi…