Eli Appleboim
In this paper we introduce two theories of finite type invariants for framed links with fixed linking matrix. We show that these thepries are related to the theory of Vassiliev invariants of framed links. We also study the corresponding spaces of ``chord diagrams''.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In the 1950's Milnor defined a family of higher order invariants generalizing the linking number. Even the first of these new invariants, the triple linking number, has received and fruitful study since its inception. In the case that $L$ has vanishing pairwise linking numbers, this triple linking number gives an integer valued invariant. When the linking numbers fail to vanish, this invariant is only well-defined modulo their greatest common divisor. In recent work Davis-Nagel-Orson-Powell produce a single invariant called the total triple linking number refining the triple linking number and…
We review a braid theoretic self-linking number formula and study its applications.
The triple linking number of an oriented surface link was defined as an analogical notion of the linking number of a classical link. We consider a certain $m$-component $T^2$-link ($m \geq 3$) determined from two commutative pure $m$-braids $a$ and $b$. We present the triple linking number of such a $T^2$-link, by using the linking numbers of the closures of $a$ and $b$. This gives a lower bound of the triple point number. In some cases, we can determine the triple point numbers, each of which is a multiple of four.
Triple linking numbers were defined for 3-component oriented surface-links in 4-space using signed triple points on projections in 3-space. In this paper we give an algebraic formulation using intersections of homology classes (or cup products on cohomology groups). We prove that spherical links have trivial triple linking numbers and that triple linking numbers are link homology invariants.
We explore a family of invariants obtained from linking numbers. This is a family of Kauffman finite type invariants.
Linking numbers of Montesinos links
本研究针对蒙蒂西诺斯链环的环绕数计算问题，给出Tuler关于双分支有理链环绕数结论的简洁证明，提出有理链环绕数的数值算法，并借此得到蒙蒂西诺斯链环任意两分支的环绕数。
Linking number of grid models
本文研究网格模型中随机链环环绕数的渐近性质，证明其矩为网格尺寸的多项式、奇数阶矩为零，并证明归一化环绕数的分布弱收敛。
This is a concise overview of the definitions and properties of the linking number and its higher-order generalization, Milnor invariants.
As a generalization of the linking number, we construct a set of invariant numbers for two-component handlebody-links. These numbers are elementary divisors associated with the natural homomorphism from the first homology group of a component to that of the complement of another component.
The linking number is the simplest link invariant given by Gauss; it is the first Gauss diagram formula expressed by one arrow among two circles. Proceeding the next stage, we study the second Gauss diagram formula consisting of two arrows among two circles. We call a function of this type the multiple linking number. There are two multiple linking numbers; one of them is ordinary Vassiliev invariant and the other function is surprisingly sensitive to the necessity of the second Reidemeister moves though any one-component Gauss diagram formula cannot detect the necessity.
We give a refined value group for the collection of triple linking numbers of links in the 3-sphere. Given two links with the same pairwise linking numbers we show that they have the same refined triple linking number collection if and only if the links admit homeomorphic surface systems. Moreover these two conditions hold if and only if the link exteriors are bordant over $B \mathbb{Z}^n$, and if and only if the third lower central series quotients $π/π_3$ of the link groups are isomorphic preserving meridians and longitudes. We also show that these conditions imply that the link groups have …
We establish some new relationships between Milnor invariants and Heegaard Floer homology. This includes a formula for the Milnor triple linking number from the link Floer complex, detection results for the Whitehead link and Borromean rings, and a structural property of the $d$-invariants of surgeries on certain algebraically split links.
Milnor's triple linking numbers of a link in the 3-sphere are interpreted geometrically in terms of the pattern of intersections of the Seifert surfaces of the components of the link. This generalizes the well known formula as an algebraic count of triple points when the pairwise linking numbers vanish.
In this paper, we introduce two functions such that the subtraction corresponds to the Milnor's triple linking number; the addition obtains a new integer-valued link homotopy invariant of $3$-component links. We also have found a series of integer-valued invariants derived from four terms whose sum equals the Milnor's triple linking number. We apply this structure to give invariants of $3$-bouquet graphs.
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.
Topological polymers have various topological types, and they are expressed by graphs. However, the Jones polynomial, we have a difficulty to compute it; computational time is growing exponentially with respect to the crossing number. The simplest Vassiliev invariant is the linking number and thus we will seek a next simple one is as the Milnor's triple linking number. In this paper, we introduce simple Gauss diagram formulas of Vassiliev invariants of Milnor type. These are non-torsion valued, whereas the base-point-free Milnor's triple linking number is usually torsion-valued.
Given a null-cobordant oriented framed link $L$ in a closed oriented $3$--manifold $M$, we determine those links in $M \setminus L$ which can be realized as the singular point set of a generic map $M \to \mathbb{R}^2$ that has $L$ as an oriented framed regular fiber. Then, we study the linking behavior between the singular point set and regular fibers for generic maps of $M$ into $\mathbb{R}^2$.
Conway and Gordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent two-component links is odd, and Kazakov and Korablev proved that for every spatial complete graph with arbitrary number of vertices greater than six, the sum of the linking numbers over all of the constituent two-component Hamiltonian links is even. In this paper, we show that for every spatial complete graph whose number of vertices is greater than six, the sum of the square of the linking numbers over all of the two-component Hamiltonian links is determin…
Let $n$, $q$ and $r$ be positive integers, and let $K_N^n$ be the $n$-skeleton of an $(N-1)$-simplex. We show that for $N$ sufficiently large every embedding of $K_N^n$ in $\mathbb{R}^{2n+1}$ contains a link $L_1\cup\cdots\cup L_r$ consisting of $r$ disjoint $n$-spheres, such that the linking number $link(L_i,L_j)$ is a nonzero multiple of $q$ for all $i\neq j$. This result is new in the classical case $n=1$ (graphs embedded in $\mathbb{R}^3$) as well as the higher dimensional cases $n\geq 2$; and since it implies the existence of a link $L_1\cup\cdots\cup L_r$ such that $|link(L_i,L_j)|\geq q…
The quadratic linking degree
本研究借助动机同伦理论，在代数几何中提出取值于基域Witt群的二次环绕度作为有向链环绕数的对应概念，探究其性质并给出显式计算方法，还在环面链环类似物实例上进行了验证。
We study Heegaard Floer homology and various related invariants (such as the $h$-function) for two-component L-space links with linking number zero. For such links, we explicitly describe the relationship between the $h$-function, the Sato-Levine invariant and the Casson invariant. We give a formula for the Heegaard Floer $d$-invariants of integral surgeries on two-component L-space links of linking number zero in terms of the $h$-function, generalizing a formula of Ni and Wu. As a consequence, for such links with unknotted components, we characterize L-space surgery slopes in terms of the $ν^…
A flat plumbing basket is a surface consisting a disk and finitely many bands which are contained in distinct pages of the trivial open book decomposition of $\mathbf{S}^{3}$. In this paper, we construct a Legendrian link from a flat plumbing basket, and we describe a relation among the self-linking number, the Thurston-Bennequin number and the flat plumbing basket number of the link. As a corollary, we determine the flat plumbing basket numbers of torus links.
We prove that two closed oriented 3-manifolds have isomorphic quintuplets (homology, space of spin structures, linking pairing, cohomology rings, Rochlin function) if, and only if, they belong to the same class of a certain surgery equivalence relation introduced by Goussarov and Habiro.
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 dimensions. In particular, we prove the existence of links of the following types: r-component links, with the linking pattern of a chain, necklace or keyring; 2-component links with linking number at least lambda in absolute value; and 2-component links with linking number a non-…
We introduce a quandle invariant of classical and virtual links, denoted $Q_{tc} (L)$. This quandle has the property that $Q_{tc} (L) \cong Q_{tc} (L')$ if and only if the components of $L$ and $L'$ can be indexed in such a way that $L=K_1 \cup \dots \cup K_μ$, $L'=K'_1 \cup \dots \cup K'_μ$ and for each index $i$, there is a multiplier $ε_i \in \{-1,1\}$ that connects virtual linking numbers over $K_i$ in $L$ to virtual linking numbers over $K'_i$ in $L'$: $\ell_{j/i}(K_i,K_j)= ε_i \ell_{j/i}(K'_i,K'_j)$ for all $j \neq i$. We also extend to virtual links a classical theorem of Chen, which re…
A triple torsion linking form and 3-manifolds in $S^4$
本文在有理同调3-球面上引入三重环绕形式，并利用Matsumoto的高阶相交形式将其与单连通4-流形边界联系起来，从而为$S^4$中的有理同调球面建立新的嵌入障碍，推广了Hantzsche的经典定理。
We study intrinsically linked graphs where we require that every embedding of the graph contains not just a non-split link, but a link that satisfies some additional property. Examples of properties we address in this paper are: a two component link with lk(A,L) = k2^r, k not 0, a non-split n-component link where all linking numbers are even, or an n-component link with components L, A_i where lk(L,A_i) = 3k, k not 0. Links with other properties are considered as well. For a given property, we prove that every embedding of a certain complete graph contains a link with that property. The size o…
The paper concerns the tree invariants of string links, introduced by Kravchenko and Polyak and closely related to the classical Milnor linking numbers also known as $\barμ$--invariants. We prove that, analogously as for $\barμ$--invariants, certain residue classes of tree invariants yield link homotopy invariants of closed links. The proof is arrow diagramatic and provides a more geometric insight into the indeterminacy through certain tree stacking operations. Further, we show that the indeterminacy of tree invariants is consistent with the original Milnor's indeterminacy. For practical purp…
The Thurston-Bennequin invariant provides one notion of self-linking for any homologically-trivial Legendrian curve in a contact three-manifold. Here we discuss related analytic notions of self-linking for Legendrian knots in Euclidean space. Our definition is based upon a reformulation of the elementary Gauss linking integral and is motivated by ideas from supersymmetric gauge theory. We recover the Thurston-Bennequin invariant as a special case.
We prove that satellite operations that satisfy a certain positivity condition and have winding number other than one are not homomorphisms. The argument uses the $d$-invariants of branched covers. In the process, we prove a technical result relating $d$-invariants and the Torelli group which may be of independent interest.
Relative self-linking and linking "numbers" for pairs of knots in oriented 3-manifolds are defined in terms of intersection invariants of immersed surfaces in 4-manifolds. The resulting concordance invariants generalize the usual homological notion of linking by taking into account the fundamental group of the ambient manifold and often map onto infinitely generated groups. The knot invariants generalize the cyclic (type 1) invariants of Kirk and Livingston and when taken with respect to certain preferred knots, called spherical knots, relative self-linking numbers are characterized geometrica…
Combinatorial two-player games have recently been applied to knot theory. Examples of this include the Knotting-Unknotting Game and the Region Unknotting Game, both of which are played on knot shadows. These are turn-based games played by two players, where each player has a separate goal to achieve in order to win the game. In this paper, we introduce the Linking-Unlinking Game which is played on two-component link shadows. We then present winning strategies for the Linking-Unlinking Game played on all shadows of two-component rational tangle closures and played on a large family of general t…
We derive an integral formula for the linking number of two submanifolds of the n-sphere S^n, of the product S^n x R^m, and of other manifolds which appear as "nice" hypersurfaces in Euclidean space. The formulas are geometrically meaningful in that they are invariant under the action of the special orthogonal group on the ambient space.
To each three-component link in the 3-sphere, we associate a geometrically natural characteristic map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its characteristic map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two-component link in 3-space is the degree of its associated Gauss map from the 2-torus to the 2-sphere. When the pairwise linking numbers are all zero, we giv…
We compute the linking number of two modular knots in the space $\text{PSL}(2, \mathbb{Z})\backslash\text{PSL}(2, \mathbb{R})$ with the trefoil filled in, which answers a question posed by Ghys in 2007. This computation is realized through the correspondence between modular links and Lorenz links, and can be thought of as an intersection number involving Conway topographs. We compare this to a second formula for the linking number of Lorenz links, which was proven by Stephen F. Kennedy in 1994.
In the note we study Legendrian and transverse knots in rationally null-homologous knot types. In particular we generalize the standard definitions of self-linking number, Thurston-Bennequin invariant and rotation number. We then prove a version of Bennequin's inequality for these knots and classify precisely when the Bennequin bound is sharp for fibered knot types. Finally we study rational unknots and show they are weakly Legendrian and transversely simple. This version of the paper corrects the definition of rational self-linking number in the previous and published version of the paper. Wi…
We introduce a notion of intrinsic linking and knotting for virtual spatial graphs. Our theory gives two filtrations of the set of all graphs, allowing us to measure, in a sense, how intrinsically linked or knotted a graph is; we show that these filtrations are descending and non-terminating. We also provide several examples of intrinsically virtually linked and knotted graphs. As a byproduct, we introduce the {\it virtual unknotting number} of a knot, and show that any knot with non-trivial Jones polynomial has virtual unknotting number at least 2.
We study the space of "link maps": the space of maps of a disjoint union of compact, closed manifolds P_1, . . ., P_k into a manifold N whose images are pairwise disjoint. We apply the manifold calculus of functors developed by Goodwillie and Weiss to study the difference between it and its linear and quadratic approximations. We identify an appropriate generalization of the linking number as the geometric object which measures the difference between the space of link maps and its linear approximation. Our analysis of the difference between link maps and its quadratic approximation connects wi…
In the course of our work on low-volume hyperbolic 3-manifolds, we came upon a linking problem for horoball necklaces in $\mathbb{H}^3$. A horoball necklace is a collection of sequentially tangent beards (i.e. spheres) with disjoint interiors lying on a flat table (i.e. a plane) such that each bead is of diameter at most one and is tangent to the table. In this note, we analyze the possible configurations of an 8-bead necklace linking around two other diameter-one spheres on the table. We show that all the beads are forced to have diameter one, the two linked spheres are tangent, and that each…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。