Blake Mellor
We extend the notion of intersection graphs for knots in the theory of finite type invariants to string links. We use our definition to develop weight systems for string links via the adjacency matrix of the intersection graph, and show that these weight systems are related to the weight systems induced by the Conway and Homfly polynomials.
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By using the equivariant localization formula of toric varieties. We prove the vanishing of the Witten genus of some string complete intersections in smooth toric varieties.
Twisted links are a generalization of classical links and correspond to stably equivalence classes of links in thickened surfaces. In this paper we introduce twisted intersection colorings of a diagram and construct two invariants of a twisted link using such colorings. As an application, we show that there exist infinitely many pairs of twisted links such that for each pair the two twisted links are not equivalent but their double coverings are equivalent. We also introduce a method of constructing a pair of twisted links whose double coverings are equivalent.
In a previous paper, we defined an operation $μ$ that generalizes Turaev's cobracket for loops on a surface. We showed that, in contrast to the cobracket, this operation gives a formula for the minimum number of self-intersections of a loop in a given free homotopy class. In this paper we consider the corresponding question for virtual strings. We show that $μ$ gives a bound on the minimal self-intersection number of a virtual string which is stronger than a bound given by Turaev's virtual string cobracket. We use Turaev's based matrices to describe strings $α$ such that $μ$ gives a formula fo…
It is well-known that all 2-knots are slice. Are all 2-links slice? This is an outstanding open question. In this paper we prove the following: For any 2-component 2-link (J,K)in the 4-sphere which bounds the 5-ball B^5, there is an embedded disc 2-disc D^2_J (respectively, D^2_K) in B^5 with the following properties: J (respectively K) bounds D^2_J (respectively, D^2_K). D^2_J and D^2_K intersect transversely. the intersection of D^2_J and D^2_K in D^2_J (respectively, D^2_K) is a trivial 1-knot.
On Intersection of Character Varieties Associated to Dehn Fillings of the Whitehead Link and Its Generalization
本文以怀特海德链为示例，研究双曲2桥链补空间经不同Dehn填充后得到的广义曲线在典范特征簇曲面上的相交情况，利用扭曲Alexander多项式分类对应可约特征的交点，探讨其拓扑意义并推广至其他2桥链。
Cobordism of virtual string links on $n$ strands is a combinatorial generalization of link cobordism. There exists a bijection between virtual string links up to cobordisms and elements of the group $\mathbb{Z}^{n(n-1)}$. This paper also shows that virtual string links up to unwelded equivalence are classified by those groups. Finally, the related theory of welded string link cobordism is defined herein and shown to be trivial for string links with one component.
Two string links are equivalent up to $2n$-moves and link-homotopy if and only if their all Milnor link-homotopy invariants are congruent modulo $n$. Moreover, the set of the equivalence classes forms a finite group generated by elements of order $n$. The classification induces that if two string links are equivalent up to $2n$-moves for every $n>0$, then they are link-homotopic.
Habegger and Lin gave a classification of the link-hmotopy classes of links as the link-homotopy classes of string links modulo the actions of conjugations and partial conjugations for string links. In this paper, we calculated the actions of the partial conjugations and the conjugations explicitly for 4- and 5-component string links which gave classifications (presentations) of the link-homotopy classes of 4- and 5-component links. As an application, we can run Habegger and Lin's algorithm which determines whether given two links are link-homotopoic or not for 4- and 5-component links.
We show that the Casson knot invariant, linking number and Milnor's triple linking number, together with a certain 2-string link invariant $V_2$, are necessary and sufficient to express any string link Vassiliev invariant of order two. Explicit combinatorial formulas are given for these invariants. This result is applied to the theory of claspers for string links.
In this paper we investigate the virtual string links via a probabilistic interpretation. This representation can be used to distinguish some virtual string links from classical string links. In order to study the algebraic structure behind this probabilistic interpretation we introduce the notion of virtual flat biquandle. The cocycle invariants associated with virtual flat biquandle is discussed.
We show that the map obtained by viewing a geometric (ie. representative) braid as a string link induces an isomorphism of the n-strand braid group onto the group of units of the n-strand string link monoid.
In this paper, we give a complete set of finite type string link invariants of degree <5. In addition to Milnor invariants, these include several string link invariants constructed by evaluating knot invariants on certain closure of (cabled) string links. We show that finite type invariants classify string links up to C_k-moves for k<6, which proves, at low degree, a conjecture due to Goussarov and Habiro. We also give a similar characterization of finite type concordance invariants of degree <6.
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.
In a previous paper, the authors proved that Milnor link-homotopy invariants modulo $n$ classify classical string links up to $2n$-move and link-homotopy. As analogues to the welded case, in terms of Milnor invariants, we give here two classifications of welded string links up to $2n$-move and self-crossing virtualization, and up to $V^{n}$-move and self-crossing virtualization, respectively.
This paper describes the relationship between the first non-vanishing Milnor invariants of a classical link and the intersection invariant of a twisted Whitney tower. This is a certain 2-complex in the 4-ball, built from immersed disks bounded by the given link in the 3-sphere together with finitely many `layers' of Whitney disks. The intersection invariant is a higher-order generalization of the intersection number between two immersed disks in the 4-ball, well known to give the linking number of the link on the boundary, which measures intersections among the Whitney disks and the disks boun…
We show that the exterior powers of the matrix valued random walk invariant of string links, introduced by Lin, Tian, and Wang, are isomorphic to the graded components of the tangle functor associated to the Alexander Polynomial by Ohtsuki divided by the zero graded invariant of the functor. Several resulting properties of these representations of the string link monoids are discussed.
We extend knot Floer homology to string links in D^{2} \times I and to d-based links in arbitrary three manifolds, without any hypothesis on the null-homology of the components. As for knot Floer homology we obtain a description of the Euler characteristic of the resulting homology groups (in D^{2} \times I) in terms of the torsion of the string link. Additionally, a state summation approach is described using the equivalent of Kauffman states. Furthermore, we examine the situtation for braids, prove that for alternating string links the Euler characteristic determines the homology, and develo…
In previous work, the author defined the intersection graph of a chord diagram associated with string links (as in the theory of finite type invariants). In this paper, we classify the trees which can be obtained as intersection graphs of string link diagrams.
In this paper we study further when tangles embed into the unknot, the unlink or a split link. In particular, we study obstructions to these properties through geometric characterizations, tangle sums and colorings. As an application we determine when each prime 2-string tangle with up to seven crossings embeds into the unknot, the unlink or a split link.
We study the Goussarov-Habiro finite type invariants theory for framed string links in homology balls. Their degree 1 invariants are computed: they are given by Milnor's triple linking numbers, the mod 2 reduction of the Sato-Levine invariant, Arf and Rochlin's $μ$ invariant. These invariants are seen to be naturally related to invariants of homology cylinders through the so-called Milnor-Johnson correspondence: in particular, an analogue of the Birman-Craggs homomorphism for string links is computed. The relation with Vassiliev theory is studied.
We employ Hirzebruch-type invariants obtained from iterated p-covers to investigate concordance of links and string links. We show that the invariants naturally give various group homomorphisms of the string link concordance group into L-groups over number fields. We also obtain homomorphisms of successive quotients of the Cochran-Orr-Teichner filtration. As an application we show that the kernel of Harvey's $ρ_n$-invariant is large enough to contain a subgroup with infinite rank abelianization, modulo local knots. As another application, we show that recently discovered nontrivial 2-torsion e…
A prime decomposition theorem for string links in a thickened surface
本文证明了加厚曲面中非辫弦链可唯一分解为若干素弦链的连通和，分解在因子置换意义下唯一。
We publish a table of primitive finite-type invariants of order less than or equal to six, for knots of ten or fewer crossings. We note certain mod-2 congruences, one of which leads to a chirality criterion in the Alexander polynomial. We state a computational result on mod-2 finite-type invariants of 2-strand string links.
Ribbon 2-knotted objects are locally flat embeddings of surfaces in 4-space which bound immersed 3-manifolds with only ribbon singularities. They appear as topological realizations of welded knotted objects, which is a natural quotient of virtual knot theory. In this paper we consider ribbon tubes and ribbon torus-links, which are natural analogues of string links and links, respectively. We show how ribbon tubes naturally act on the reduced free group, and how this action classifies ribbon tubes up to link-homotopy, that is when allowing each component to cross itself. At the combinatorial le…
It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called "splice type singularities", which generalize Brieskorn complete intersections. Second, these arise as universal abelian covers of a class of normal surface singularities with Q-homology sphere links, called "splice-quotient singularities". According to the Main Theorem, splice-quo…
The Gassner representation of the pure braid group to $GL_n(Z[Z^n])$ can be extended to give a representation of the concordance group of $n$-strand string links to $GL_n(F)$, where $F$ is the field of quotients of $\zz[\zz^n]$, $ F = Q(t_1,...,t_n)$; this was first observed by Le Dimet. Here we present simple new perspectives on its definition, its computation, and its applications to obtain both general results and interesting family of examples.
We show that the Artin representation on concordance classes of string links induces a well-defined epimorphism modulo order n twisted Whitney tower concordance, and that the kernel of this map is generated by band sums of iterated Bing-doubles of any string knot with nonzero Arf invariant. We also continue J. Levine's work [20, 21, 22] comparing two filtrations of the group of homology cobordism classes of 3-dimensional homology cylinders, one defined in terms of an Artin-type representation (the Johnson filtration) and one defined using clasper surgery (the Goussarov-Habiro filtration). In p…
We consider knotted annuli in 4-space, called 2-string-links, which are knotted surfaces in codimension two that are naturally related, via closure operations, to both 2-links and 2-torus links. We classify 2-string-links up to link-homotopy by means of a 4-dimensional version of Milnor invariants. The key to our proof is that any 2-string link is link-homotopic to a ribbon one; this allows to use the homotopy classification obtained in the ribbon case by P. Bellingeri and the authors. Along the way, we give a Roseman-type result for immersed surfaces in 4-space. We also discuss the case of ri…
The set SL(n) of n-string links has a monoid structure, given by the stacking product. When considered up to concordance, SL(n) becomes a group, which is known to be abelian only if n=1. In this paper, we consider two families of equivalence relations which endow SL(n) with a group structure, namely the C_k-equivalence introduced by Habiro in connection with finite type invariants theory, and the C_k-concordance, which is generated by C_k-equivalence and concordance. We investigate under which condition these groups are abelian, and give applications to finite type invariants.
On finite type invariants of welded string links and ribbon tubes
本文利用焊接对象研究带状纽结曲面的有限型不变量，分析w_k等价下焊接弦链环的性质，证明低阶时该等价关系可刻画有限型不变量信息，并探究其代数结构，结果对带状纽结曲面有直接推论。
We exhibit an explicit delooping of the space of string links with $m$ components in the hypercube $I^{n}$, for $n\geq 4$ and $m\geq 1$.
A virtual string is a scheme of self-intersections of a closed curve on a surface. We introduce virtual strings and study their geometric properties and homotopy invariants. We also discuss connections between virtual strings, Gauss words, and virtual knots.
Budney recently constructed an operad that encodes splicing of knots. He further showed that the space of (long) knots is generated over this operad by the space of torus knots and hyperbolic knots, thus generalizing the satellite decomposition of knots from isotopy classes to the level of the space of knots. Infection by string links is a generalization of splicing from knots to links. We construct a colored operad that encodes string link infection. We prove that a certain subspace of the space of 2-component string links is generated over a suboperad of our operad by its subspace of prime l…
While the topological types of {normal} surface singularities with homology sphere link have been classified, forming a rich class, until recently little was known about the possible analytic structures. We proved in [Geom. Topol. 9(2005) 699-755] that many of them can be realized as complete intersection singularities of "splice type", generalizing Brieskorn type. We show that a normal singularity with homology sphere link is of splice type if and only if some naturally occurring knots in the singularity link are themselves links of hypersurface sections of the singular point. The Casson Inva…
Let S^3_i be a 3-sphere embedded in the 5-sphere S^5 (i=1,2). Let S^3_1 and S^3_2 intersect transversely. Then the intersection C of S^3_1 and S^3_2 is a disjoint collection of circles. Thus we obtain a pair of 1-links, C in S^3_i (i=1,2), and a pair of 3-knots, S^3_i in S^5 (i=1,2). Conversely let (L_1,L_2) be a pair of 1-links and (X_1,X_2) be a pair of 3-knots. It is natural to ask whether the pair of 1-links (L_1,L_2) is obtained as the intersection of the 3-knots X_1 and X_2 as above. We give a complete answer to this question. Our answer gives a new geometric meaning of the Arf invariant…
Koschorke introduced a map from the space of closed $n$-component links to the ordered configuration space of $n$-tuples of points in $\mathbb{R}^3$, and conjectured that this map separates homotopy links. The purpose of this paper is to construct an analogous map for string links, and to prove (1) this map in fact separates homotopy string links, and (2) Koschorke's original map factors through the map constructed here together with an analogue of Markov's closure map defined on the level of certain function spaces.
Configuration space integrals have in recent years been used for studying the cohomology of spaces of (string) knots and links in $\mathbb{R}^n$ for $n>3$ since they provide a map from a certain differential algebra of diagrams to the deRham complex of differential forms on the spaces of knots and links. We refine this construction so that it now applies to the space of homotopy string links -- the space of smooth maps of some number of copies of $\mathbb{R}$ in $\mathbb{R}^n$ with fixed behavior outside a compact set and such that the images of the copies of $\R$ are disjoint -- even for $n=3…
In previous work, we defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we look at the case when this graph is a tree, and we show that in many cases these trees determine the chord diagram (modulo the usual 1-term and 4-term relations).
In this article, we define and study the total Milnor invariant and the infinitesimal Morita-Milnor homomorphism as punctured disk analogues of the total Johnson map and the infinitesimal Morita homomorphism studied by Kawazumi and Massuyeau in the case of surface of positive genus with one boundary component.
Operadic actions on long knots and 2-string links
本文将2-弦链空间实现为标色算子$\mathcal{SCL}$上的自由代数，扩展了前人关于商空间和长结自由性的结果，并从代数和拓扑角度细化了弦链同痕类幺半群的定理。
We introduce the (general) homotopy groups of spheres as link invariants for Brunnian-type links through the investigations on the intersection subgroup of the normal closures of the meridians of strongly nonsplittable links. The homotopy groups measure the difference between the intersection subgroup and symmetric commutator subgroup of the normal closures of the meridians and give the invariants of the links obtained in this way. Moreover the higher homotopy-group invariants can produce some links that could not be detected by the Milnor invariants. Furthermore all homotopy groups of spheres…
In this paper, we first endow the set of ribbon string links (up to isotopy) with a structure of a cyclic and of a cocyclic set. Next, we relate these (co)cyclic sets with those associated with the coend of a ribbon category. The relationship is given by the universal quantum invariants à la Reshetikhin-Turaev.
Representation stability of string links and manifold links
本文证明弦链与流形链空间的理性上同调具有表示稳定性，给出生成元数量上界，并得到理性同伦群的显式生成界。
We refine the intersection product in homology to an equivariant setting, which unifies several known constructions. As an application, we give a common generalisation of the Chas-Sullivan string product on a manifold and the Chataur-Menichi string product on the classifying space by defining a string product on the Borel construction of a manifold. We prove a vanishing result which enables us to define a secondary product. The secondary product is then used to construct secondary versions of the Chataur-Menichi string product, and the equivariant intersection product in the Borel equivariant …
Witten genera of complete intersections
研究完全交集的Witten genera在特定流形中的消失性质，并通过Fano流形验证Stolz猜想
The Kontsevich invariant and the action of the Grothendieck--Teichmüller group on $2$-component string links
本文研究了 Kontsevich 不变量对关联子选取的依赖性，并展示了 Grothendieck--Teichmüller 群对 2-组件字符串链代数作用的非平凡性。
An exact sequence for generalized string links over surfaces
本文将Goldberg的结果推广至亏格大于等于1的闭连通可定向曲面上的广义弦链，研究了其至多相差链同伦的性质。
Bott and Taubes constructed knot invariants by integrating differential forms along the fiber of a bundle over the space of knots, generalizing the Gauss linking integral. Their techniques were later used to construct real cohomology classes in spaces of knots and links in higher-dimensional Euclidean spaces. In previous work, we constructed cohomology classes in knot spaces with arbitrary coefficients by integrating via a Pontrjagin--Thom construction. We carry out a similar construction over the space of string links, but with a refinement in which configuration spaces are glued together acc…
We discuss an action of the Grothendieck-Teichmüller proalgebraic group on the linear span of proalgebraic tangles, oriented tangles completed by a filtration of Vassiliev. The action yields a motivic structure on tangles. We derive distinguished properties of the action particularly on proalgebraic string links and on proalgebraic knots which can not be observed in the action on proalgebraic braids. By exploiting the properties, we explicitly calculate the inverse image of the trivial (the chordless) chord diagram under the Kontsevich isomorphism.
Let M be a closed Riemannian manifold. We extend the product of Goresky-Hingston, on the cohomology of the free loop space of M relative to the constant loops, to a nonrelative product. It is graded associative and commutative, and compatible with the length filtration on the loop space, like the original product. We prove the following new geometric property of the dual homology coproduct: the nonvanishing of the k-th iterate of the coproduct on a homology class ensures the existence of a loop with a (k+1)-fold self-intersection in every representative of the class. For spheres and projective…
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