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In 2014, Vaughan Jones developed a method to produce links from elements of Thompson’s group $F$, and showed that all links arise this way. He also introduced a subgroup $\overrightarrow{F}$ of $F$ and a method to produce oriented links from elements of this subgroup. In 2018, Valeriano Aiello showed that all oriented links arise from this construction. We classify exactly those regular isotopy classes of links that arise from $F$, as well as exactly those regular isotopy classes of oriented links that arise from $\overrightarrow{F}$, answering a question asked by Jones in 2018.

In [Ilyutko and Safina, Graph-links: nonrealizability, orientation, and Jones polynomial, J. Math. Sc.214(5) (2016) 632–664], the first named author defined the notion of an oriented graph-link and constructed a writhe number of a vertex of an oriented graph-link, which equals the “real” writhe number of a crossing in the realizable case. As a result the Jones polynomial was defined for oriented graph-links and the first example of a non-realizable graph-link with more than one component was found. Despite the fact that all necessary definitions were given the authors did not define the notion of linking number for graph-links. In this paper, we are eliminating this deficiency. Moreover, we classify all graph-links having a representative with less than $4$ vertices.

The paper relates the 4-fold symmetric quandle homotopy (cocycle) invariants to topological objects. We show that the 4-fold symmetric quandle homotopy invariants are at least as powerful as the Dijkgraaf–Witten invariants. As an application, for an odd prime p, we show that the quandle cocycle invariant of a link in S³ constructed by the Mochizuki 3-cocycle is equivalent to the Dijkgraaf–Witten invariant with respect to ℤ/pℤ of the double covering of S³ branched along the link. We also reconstruct the Chern–Simons invariant of closed 3-manifolds as a quandle cocycle invariant via the extended Bloch group, in analogy to [A. Inoue and Y. Kabaya, Quandle homology and complex volume, preprint(2010), arXiv:math/1012.2923].

Two link diagrams on compact surfaces are strongly equivalent if they are related by Reidemeister moves and orientation preserving homeomorphisms of the surfaces. They are stably equivalent if they are related by the two previous operations and adding or removing handles. Turaev and Turner constructed a link homology for each stable equivalence class by applying an unoriented topological quantum field theory (TQFT) to a geometric chain complex similar to Bar-Natan's one. In this paper, by using an unoriented homotopy quantum field theory (HQFT), we construct a link homology for each strong equivalence class. Moreover, our homology yields an invariant of links in the oriented I-bundle of a compact surface.

Analogues of Iwasawa invariants in the context of 3-dimensional topology have been studied by M. Morishita and others. In this paper, following the dictionary of arithmetic topology, we formulate an analogue of Kida’s formula on $\lambda$-invariants in a $p$-extension of ${\mathbb{Z}}_p$-fields for 3-manifolds. The proof is given in a parallel manner to Iwasawa’s second proof, with use of $p$-adic representations of a finite group. In the course of our arguments, we introduce the notion of a branched ${\mathbb{Z}}_p$-cover as an inverse system of cyclic branched $p$-covers of 3-manifolds, generalize the Iwasawa type formula, and compute the Tate cohomology of 2-cycles explicitly.

We extend the Gordon–Litherland pairing to links in thickened surfaces, and use it to define signature, determinant and nullity invariants for links that bound (unoriented) spanning surfaces. The invariants are seen to depend only on the $S^{\ast }$-equivalence class of the spanning surface. We prove a duality result relating the invariants from one $S^{\ast }$-equivalence class of spanning surfaces to the restricted invariants of the other. Using Kuperberg’s theorem, these invariants give rise to well-defined invariants of checkerboard colorable virtual links. The determinants can be applied to determine the minimal support genus of a checkerboard colorable virtual link. The duality result leads to a simple algorithm for computing the invariants from the Tait graph associated to a checkerboard coloring. We show these invariants simultaneously generalize the combinatorial invariants defined by Im, Lee and Lee, and those defined by Boden, Chrisman and Gaudreau for almost classical links. We examine the behavior of the invariants under orientation reversal, mirror symmetry and crossing change. We give a 4-dimensional interpretation of the Gordon–Litherland pairing by relating it to the intersection form on the relative homology of certain double branched covers. This correspondence is made explicit through the use of virtual linking matrices associated to (virtual) spanning surfaces and their associated (virtual) Kirby diagrams.

Using an argument from statistical mechanics, V. Jones has given a new method for constructing pairs of links with identical skein polynomials. We give a more general construction and use it to provide a simple proof of a theorem of Traczyk's involving rotors of links. Several examples are given of pairs of knots with coincident skein polynomials and small crossing number.

Subcube embeddability of the hypercube can be enhanced by introducing an additional dimension. A set of new dimensions, characterized by the Hamming distance between the pairs of nodes it connects, is introduced using a measure defined as the magnitude of a dimension. An enumeration of subcubes of various size is presented for a dimension parameterized by its magnitude. It is shown that the maximum number of subcubes for a Q_n can only be attained when the magnitude of dimension is n-1 or n. It is further shown that the latter two dimensions can optimally increase the number of subcubes among all possible choices.

This paper is an exposition of the relationship between Witten’s Chern–Simons functional integral and the theory of Vassiliev invariants of knots and links in three-dimensional space. We conceptualize the functional integral in terms of equivalence classes of functionals of gauge fields and we do not use measure theory. This approach makes it possible to discuss the mathematics intrinsic to the functional integral rigorously and without functional integration. Applications to loop quantum gravity are discussed.

A topological invariant, analogous to the linking number as defined in knot theory, is defined for pairs of digital closed paths of ℤ³. This kind of invariant is very useful for proofs which involve homotopy classes of digital paths. Indeed, it can be used, for example, in order to state the connection between the tunnels in an object and the ones in its complement. Even if its definition is not as immediate as in the continuous case it has the good property that it is immediately computable from the coordinates of the voxels of the paths with no need of a regular projection. The aim of this paper is to state and prove that the linking number has the same property as its continuous analogue: it is invariant under any homotopic deformation of one of the two paths in the complement of the other.

Open Source Software (OSS) community has attracted a large number of distributed developers to work together, e.g. reporting and discussing issues as well as submitting and reviewing code. OSS developers create links among development units (e.g. issues and pull requests in GitHub), share their opinions and promote the resolution of development units. Although previous work has examined the role of links in recommending high-priority tasks and reducing resource waste, the understanding of the actual usage of links in practice is still limited. To address the research gap, we conduct an empirical study based on the 5W1H model and data mining from five popular OSS projects on GitHub. We find that links originating from a PR are more common than the other three types of links, and links are more frequently created in Documentation. We also find that average duration between development units’ create time in a link is half a year. We observed that link behaviors are very complex and the duration of link increases with the complexity of link structure. We also observe that the reasons of link are very different, especially in P–P and I–I. Finally, future works are discussed in conclusion.

Extending and reproving a recent result of D. Krebes, we give obstructions to the embedding of a tangle in a link.

A double-torus knot is a knot embedded in a genus two Heegaard surface in S³. We consider double-torus knots L such that is connected, and consider fibred knots in various classes.

For an orientation preserving homeomorphism φ of the disk into itself, a suspension of a finite union of periodic orbits P of φ represents a link type in the 3-sphere S³. Let φ be a C¹ diffeomorphism, and p a hyperbolic fixed point of φ with a homoclinic point. If all the homoclinic points for p are transeverse, then for infinitely many n>0, φⁿ induces all link types, that is, for each link type L in S³, there exists a finite union of periodic orbits of φⁿ such that a suspension of of φⁿ represents L.

Let φ:D²→D² be an orientation preserving homeomorphism of the disk into itself, and Φ= {φ_t}_0≤t≤1 an isotopy with φ₀=id_D² and φ₁=φ. Then for a finite union of periodic orbits P of φ, the set is a link in D²×S¹. We say that φ induces all link types (for Φ) if there exists a homeomorphism h of D²×S¹ into a standardly embedded solid torus in the 3-sphere S³ such that any link L in S³ can be realized by a finite union of periodic orbits P_L of φ so that L and are equivalent. We will show that the Smale horseshoe and its second power do not induce all link types, but its third power does induce all link types.

We deal with the properties of derivative of Jones polynomial for algebraically split links. In particularly, the properties of ν_n(L) and ϕ_n(L) are discussed.

Examples of non-trivial 2- and 3-component links are given that cannot be distinguished from the corresponding unlink by means of the Jones polynomial.

We consider Dehn surgeries on some non-simple 2-component links in S³ which yield S³. As corollary we give a class of 2-component links which are determined by their complements.

We shall study several circles in the 3-sphere called a link which has "high" splitness properties. We offer several kind of those links and study relations among them. Alexander polynomial, Conway polynomial and Milnor μ and invariants did not work for those links as vanishing cause of high splitness. We use higher order elementary ideals to distinguish those links.

In a recent paper [8], Xiao-Song Lin gave an example of a finite type invariant of links up to link homotopy that is not simply a polynomial in the pairwise linking numbers. Here we present a reformulation of the problem of finding such polynomials using the primary geometric obstruction homomorphism, previously used to study realizability of link group automorphisms by link homotopies. Using this reformulation, we generalize Lin's results to k-trivial links (links that become homotopically trivial when any k components are deleted). Our approach also gives a method for finding torsion finite type link homotopy invariants within "linking classes," generalizing an idea explored earlier in [1] and [10], and yielding torsion invariants within linking classes that are different from Milnor's invariants in their original indeterminacy.