Narrow Results

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.

For a virtual knot $K$ and an integer $r$ with $r\ge 2$, we introduce a method of constructing an $r$-component virtual link $L(K;r)$, which we call the $r$-multiplexing of $K$. Every invariant of $L(K;r)$ is an invariant of $K$. We give a way of calculating three kinds of invariants of $L(K;r)$ using invariants of $K$. As an application of our method, we also show that Manturov’s virtual $n$-colorings for $K$ can be interpreted as certain classical $n$-colorings for $L(K;2)$.

In this paper we investigate an interpolating gauge-fixing procedure in (4l + 3)-dimensional Abelian Chern–Simons theory. We show that this interpolating gauge is related to the covariant gauge in a constant anisotropic metric. We compute the corresponding propagators involved in various expressions of the linking number in various gauges. We comment on the geometric interpretations of these expressions, clarifying how to pass from one interpretation to another.

Based on the Bogomol'nyi self-dual equation in the Abelian Chern–Simons Higgs model, we find a self-dual topological term which was ignored all the time in the Bogomol'nyi self-duality equation due to the improper decomposition of the complex Higgs field. We also present a generalized self-dual equation for the first time which unified the topological term and nontopological term in (2+1) space–time. It is shown that the self-dual vortex just arise from the symmetric phase of the Higgs field ϕ = 0, this provides a deep insight into the spontaneous symmetry breaking mechanism. When generalizing this model into (3+1) space–time, we show that the topological number on a family of knotted vortex are just the sum of linking number and self-linking number. We also find this topological charge is conserved during the splitting, the mergence and the intersection processes of knotted vortex lines.

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.

An autonomous four-dimensional dynamical system is investigated through a topological analysis. This system generates a chaotic attractor for the range of control parameters studied and we determine the organization of the unstable periodic orbits (UPOs) associated with the chaotic attractor. Surrogate UPOs were found in the four-dimensional phase space and pairs of these orbits were embedded in three-dimensions using Locally Linear Embedding. This is a dimensionality reduction technique recently developed in the machine learning community. Embedding pairs of orbits allows the computation of their linking numbers, a topological invariant. A table of linking numbers was computed for a range of control parameter values which shows that the organization of the UPOs is consistent with that of a Lorenz-type branched manifold with rotation symmetry.

The maximum of the linking number between two lattice polygons of lengths n₁, n₂ (with n₁ ≤ n₂) is proven to be the order of n₁ (n₂)^⅓. This result is generalized to smooth links of unit thickness. The result also implies that the writhe of a lattice knot K of length n is at most 26 n^4/3/π. In the second half of the paper examples are given to show that linking numbers of order n₁ (n₂)^⅓ can be obtained when . When , it is further shown that the maximum of the linking number between these two polygons is bounded by for some constant c > 0. Finally the maximal total linking number of lattice links with more than 2 components is generalized to k components.

The formula Lk = Wr + Tw, which expresses the linking number of two curves that bound a ribbon as a sum of two terms, has particularly interested biologists and was used to understand the DNA structure. The study of Legendrian curves in contact manifolds, and in particular in the Heisenberg space, is attached to some important problems in geometry, as the problem of classification of contact structures. In this work, we show the analogue formula for curves in the Heisenberg space, we relate the writhing number with the Thurston-Benequin invariant of a Legendrian curve and derive some results directly from this formula.

Let k be a non-negative integer. Then any embedding of the complete graph on 6·2^k vertices into a three-space contains a two-component link with the absolute value of its linking number greater than or equal to 2^k. Let j be a non-negative integer. Then any embedding of the complete graph on 48·2^j vertices into a three-space contains a knot with the absolute value of the second coefficient of its Conway polynomial greater than or equal to 2^2j.

For any two disjoint oriented circles embedded into the 3-dimensional real projective space, we construct a 3-dimensional configuration space and its map to the projective space such that the linking number of the circles is the half of the degree of the map. Similar interpretations are given for the linking number of cycles in a projective space of arbitrary odd dimension and the self-linking number of a zero homologous knot in the 3-dimensional projective space.

We study the quandle counting invariant for a certain family of finite quandles with trivial orbit subquandles. We show how these invariants determine the linking number of classical two-component links up to sign.

We estimate from below the number of lines meeting each of given 4 disjoint smooth closed curves in a given cyclic order in the real projective 3-space. We obtain also a similar lower bound for the number of lines meeting each of given 4 disjoint smooth closed curves in a given linear order in ℝ³. The estimates are formulated in terms of linking numbers of the curves and obtained by orienting of the corresponding configuration spaces and evaluating of their signatures. This involves a study of a surface swept by lines meeting 3 given disjoint smooth closed curves. Higher-dimensional generalizations of these results are outlined.

We consider two self-avoiding polygons (2SAPs) each of which spans a tubular sublattice of ℤ³. A pattern theorem is proved for 2SAPs, that is any proper pattern (a local configuration in the middle of a 2SAP) occurs in all but exponentially few sufficiently large 2SAPs. This pattern theorem is then used to prove that all but exponentially few sufficiently large 2SAPs are topologically linked. Moreover, we also use it to prove that the linking number Lk of an n edge 2SAP G_n satisfies lim_n→∞ℙ(|Lk(G_n)| ≥ f(n))=1 for any function . Hence the probability of a non zero linking number for a 2SAP approaches one as the size of the 2SAP goes to infinity. It is also established that, due to the tube constraint, the linking number of an n edge 2SAP grows at most linearly in n.

We define an invariant for two component spatial graphs. Although the definition of the invariant is alike a linking number, it is different from the absolute value of a linking number. We show that the invariant is not a finite type invariant.

In this paper, we give an algorithm to calculate the minimal self-intersection number of paths in a compact surface with boundary representing a given element of the free group F(x₁, x₂, …, x_n). In particular, this algorithm says whether or not a word in x₁, x₂, …, x_n is representable by a simple path. Our algorithm is simpler than similar algorithms given previously. In the case of a disk with n holes the problem is equivalent to the problem of deciding which relators can appear in an Artin n-presentation.

In this paper we provide a mathematical reconstruction of what might have been Gauss' own derivation of the linking number of 1833, providing also an alternative, explicit proof of its modern interpretation in terms of degree, signed crossings and intersection number. The reconstruction presented here is entirely based on an accurate study of Gauss' own work on terrestrial magnetism. A brief discussion of a possibly independent derivation made by Maxwell in 1867 completes this reconstruction. Since the linking number interpretations in terms of degree, signed crossings and intersection index play such an important role in modern mathematical physics, we offer a direct proof of their equivalence. Explicit examples of its interpretation in terms of oriented area are also provided.

An oriented and ordered n-component link L in the 3-sphere is said to be achiral if it is ambient isotopic to its mirror image ignoring orientations and ordering of the components. For an oriented and ordered n-component link L, let λ_L be the product of linking numbers of all 2-component sublinks in L. For n = 4m + 3, where m is a non-negative integer, we show that if L is achiral then λ_L = 0. And for n ≠ 4m + 3, we show that there exists an n-component achiral link L with λ_L ≠ 0 by using achiral embeddings of complete graphs with n vertices K_n.

For any oriented link of two components in an integral homology 3-sphere, we define an instanton Floer homology whose Euler characteristic is twice the linking number between the components of the link. We show that, for two-component links in the 3-sphere, this Floer homology does not vanish unless the link is split. We also relate our Floer homology to the Kronheimer–Mrowka instanton Floer homology for links.

Gusarov and Habiro introduced a C_m move, that is strongly related to Vassiliev invariants. In this note, we study a special kind of C_m move, called a non-self C_m move. We show that two links can be transformed into each other by a finite sequence of non-self C_m moves if and only if (1) the two links can be transformed into each other by a finite sequence of C_m moves, and (2) the knot types of corresponding components coincide.

We construct two knot invariants. The first knot invariant is a matrix constructed using linking numbers. This matrix can be represented as a two variable polynomial. The second is an invariant of flat knots and is a formal sum of flat knots obtained by smoothing pairs of crossings. This invariant can be used in conjunction with other flat invariants, forming a family of invariants. Both invariants are constructed using the parity of a crossing.