String Links Research Articles

Homotopy Brunnian links and the $\kappa $-invariant

We provide an alternative proof that Koschorke's $\kappa$-invariant is injective on the set of link homotopy classes of $n$-component homotopy Brunnian links $BLM(n)$. The existing proof (by Koschorke \cite{Koschorke97}) is based on the Pontryagin--Thom theory of framed cobordisms, whereas ours is closer in spirit to techniques based on Habegger and Lin's string links. We frame the result in the language of Fox's torus homotopy groups and the rational homotopy Lie algebra of the configuration space $\text{Conf}(n)$ of $n$ points in $\mathbb{R}^3$. It allows us to express the relevant Milnor's $\mu$--invariants as homotopy periods of $\text{Conf}(n)$.

The (n)–solvable filtration of link concordance and Milnor’s invariants

We establish several new results about both the (n)-solvable filtration, F_n^m, of the set of link concordance classes and the (n)-solvable filtration of the string link concordance group. We first establish a relationship between Milnor's invariants and links, L, with certain restrictions on the 4-manifold bounded by M_L. Using this relationship, we can relate (n)-solvability of a link (or string link) with its Milnor's invariants. Specifically, we show that if a link is (n)-solvable, then its Milnor's invariants vanish for lengths up to 2^{n+2}-1. Previously, there were no known results about the "other half" of the filtration, namely F_{n.5}^m/F_{n+1}^m. We establish the effect of the Bing doubling operator on (n)-solvability and using this, we show that F_{n.5}^m/F_{n+1}^m is nontrivial for links (and string links) with sufficiently many components. Moreover, we show that these quotients contain an infinite cyclic subgroup. We also show that links and string links modulo (1)-solvability is a nonabelian group. We show that we can relate other filtrations with Milnor's invariants. We show that if a link is n-positive, then its Milnor's invariants will also vanish for lengths up to 2^{n+2}-1. Lastly, we prove that the Grope filtration, G_n^m, of the set of link concordance classes is not the same as the (n)-solvable filtration.

Infection by string links and new structure in the knot concordance group

This paper highlights the importance of string link concordance in the understanding of knot concordance in general. The results of this paper show that there are infinitely many nontrivial knots in the groups Gn of n‐solvable knots modulo n:5‐solvable knots, for n greater than or equal to 2, which are not concordant to any knot that is obtained by two or more iterated infections of an Arf invariant zero knot by knots. This latter class accounts for nearly all previously known examples of knots in Gn , n greater than or equal to 2. In this paper we will generalize the concept of when a rational Laurent polynomial is strongly coprime to another, first introduced by Cochran, Harvey and Leidy, to include multivariable polynomials. We also prove the existence of multivariable polynomials which are strongly coprime to all single variable Laurent polynomials. From this definition of coprimality we define the derived series localized at M for a given sequence of multivariable polynomials M. From such series we obtain refinements of the n‐solvable filtration. The operation of infection by a string link is then used to demonstrate that for particular M, certain quotients of successive terms of these refined filtrations have infinite rank. 57M25; 20J05

The Milnor triple linking number of string links by cut-and-paste topology

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 according to the combinatorics of weight systems. This gluing is somewhat similar to work of Kuperberg and Thurston. We use a formula of Mellor for weight systems of Milnor invariants, and we thus recover the Milnor triple linking number for string links, which is in some sense the simplest interesting example of a class obtained by this gluing refinement of our previous methods. Along the way, we find a description of this triple linking number as a "degree" of a map from the 6-sphere to a quotient of the product of three 2-spheres.

Units of the string link monoids

We show that the map obtained by viewing a geometric (i.e. 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.

CONFIGURATION SPACE INTEGRALS AND THE COHOMOLOGY OF THE SPACE OF HOMOTOPY STRING LINKS

Configuration space integrals have been used in recent years for studying the cohomology of spaces of (string) knots and links in ℝn for n > 3 since they provide a map from a certain differential graded 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 ℝ in ℝn with fixed behavior outside a compact set and such that the images of the copies of ℝ are disjoint — even for n = 3. We further study the case n = 3 in degree zero and show that our integrals represent a universal finite type invariant of the space of classical homotopy string links. As a consequence, we deduce that Milnor invariants of string links can be written in terms of configuration space integrals.

Cosimplicial models for spaces of links

We study the spaces of string links and homotopy string links in an arbitrary manifold using multivariable manifold calculus of functors. We construct multi-cosimplicial models for both spaces and deduce certain convergence properties of the associated Bousfield-Kan homotopy and cohomology spectral sequences when the ambient manifold is a Euclidean space of dimension four or more.

Size and configuration syntheses of rigid-link mechanisms with multiple rotary actuators using the constraint force design method

This study presents a new synthetic approach for path-generating or function-generating rigid-body mechanisms in the framework of a new hybrid genetic algorithm. In spite of some advantages of synthesis methods in designing rigid-body mechanisms, some inherent issues remain in determining optimal sizes and configurations of rigid links with multiple rotary actuators. To alleviate these difficulties and limitations, we improve our previous contribution, called the constraint force design method, by parameterizing and optimizing the existence of links, the x–y locations of joints modeled by unit masses (particles), and the kind of selection between rigid and string links, using binary, integer, and binary design variables, respectively. This new genotype coding system for GA for the integer and binary phenotypes makes it possible to use a smaller number of unit masses to synthesize manifold configurations of rigid-body mechanisms. In addition, the locations and types of rotary actuators are parameterized with additional integer design variables for the synthesis of realistic rigid-body mechanisms. Furthermore, for efficient optimization of GA, a new h-GA integrated with the Sequential Quadratic Programming (SQP) optimization algorithm is also developed to optimize the locations of joints. To demonstrate the validity of the present constraint force design method, several mechanism synthesis problems are solved.

Higher-order intersections in low-dimensional topology

We show how to measure the failure of the Whitney move in dimension 4 by constructing higher-order intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with previously known link invariants such as Milnor, Sato-Levine, and Arf invariants. We also define higher-order Sato-Levine and Arf invariants and show that these invariants detect the obstructions to framing a twisted Whitney tower. Together with Milnor invariants, these higher-order invariants are shown to classify the existence of (twisted) Whitney towers of increasing order in the 4-ball. A conjecture regarding the nontriviality of the higher-order Arf invariants is formulated, and related implications for filtrations of string links and 3-dimensional homology cylinders are described.

The Stabilizer of 1 in Habegger-Lin's Action for String Links

Let $H(k)$ be the group of all homotopy classes of $k$-string links. It has been proved that $f,g\in H(k)$ have the same closure if and only if there is $\beta\in S_k(1)$ such that $\beta\cdot f=g$, where $S_k(1)$ is the stabilizer of 1 for a certain action of the group $H(2k)$ on the set $H(k)$. If $\beta\in S_k(1)$, Artin's automorphism $\overline{\beta}$, induced by $\beta$ on $RF(2k)$, the reduced free group in $2k$ generators, induces an automorphism $\overline{\overline{\beta}}\in A(RF(k))$, the group of all automorphisms of $RF(k)$ that send each generator to one of its conjugates. This can be used to compare the homotopy classes of links obtained by closing $f$ and $g$. The association $\beta\mapsto \overline{\overline{\beta}}$ is a homomorphism from $S_k(1)$ to $A(RF(k))$. In this paper we determine its kernel.

The non-triviality of the Grope filtrations of the knot and link concordance groups

We consider the Grope filtration of the classical knot concordance group that was introduced by Cochran, Orr and Teichner. Our main result is that each successive quotient in this filtration has infinite rank. We also establish the analogous result for the Grope filtration of the concordance group of string links consisting of more than one component.

Characterization of finite type string link invariants of degree <5

AbstractWe 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 closures of (cabled) string links. We show that finite type invariants classify string links up toCk-moves fork≤ 5, which proves, at low degree, a conjecture due to Goussarov and Habiro. We also give a similar classification of string links up toCk-moves and concordance fork≤ 6.

Poincaré series of Klein groups, coxeter polynomials, the Burau representation, and Milnor invariants

We obtain several formulas for the Poincare series defined by B. Kostant for Klein groups (binary polyhedral groups) and some formulas for Coxeter polynomials (characteristic polynomials of monodromy in the case of singularities). Some of these formulas—the generalized Ebeling formula, the Christoffel-Darboux identity, and the combinatorial formula—are corollaries to the well-known statements on the characteristic polynomial of a graph and are analogous to formulas for orthogonal polynomials. The ratios of Poincare series and Coxeter polynomials are represented in terms of branched continued fractions, which are q-analogs of continued fractions that arise in the theory of resolution of singularities and in the Kirby calculus. Other formulas connect the ratios of some Poincare series and Coxeter polynomials with the Burau representation and Milnor invariants of string links. The results obtained by S.M. Gusein-Zade, F. Delgado, and A. Campillo allow one to consider these facts as statements on the Poincare series of the rings of functions on the singularities of curves, which suggests the following conjecture: the ratio of the Poincare series of the rings of functions for close (in the sense of adjacency or position in a series) singularities of curves is determined by the Burau representation or by the Milnor invariants of a string link, which is an intermediate object in the transformation of the knot of one singularity into the knot of the other.

Distinguishing links up to link-homotopy by algebraic methods

In this paper we provide a complete algebraic invariant of link-homotopy, that is, an algebraic invariant that distinguishes two links if and only if they are link-homotopic. The paper establishes a connection between the “peripheral structures” approach to link-homotopy taken by Milnor, Levine and others, and the string link action approach taken by Habegger and Lin.

Self delta-equivalence for links whose Milnor’s isotopy invariants vanish

For an n n -component link, Milnor’s isotopy invariants are defined for each multi-index I = i 1 i 2 . . . i m ( i j ∈ { 1 , . . . , n } ) I=i_1i_2...i_m~(i_j\in \{1,...,n\}) . Here m m is called the length. Let r ( I ) r(I) denote the maximum number of times that any index appears in I I . It is known that Milnor invariants with r = 1 r=1 , i.e., Milnor invariants for all multi-indices I I with r ( I ) = 1 r(I)=1 , are link-homotopy invariant. N. Habegger and X. S. Lin showed that two string links are link-homotopic if and only if their Milnor invariants with r = 1 r=1 coincide. This gives us that a link in S 3 S^3 is link-homotopic to a trivial link if and only if all Milnor invariants of the link with r = 1 r=1 vanish. Although Milnor invariants with r = 2 r=2 are not link-homotopy invariants, T. Fleming and the author showed that Milnor invariants with r ≤ 2 r\leq 2 are self Δ \Delta -equivalence invariants. In this paper, we give a self Δ \Delta -equivalence classification of the set of n n -component links in S 3 S^3 whose Milnor invariants with length ≤ 2 n − 1 \leq 2n-1 and r ≤ 2 r\leq 2 vanish. As a corollary, we have that a link is self Δ \Delta -equivalent to a trivial link if and only if all Milnor invariants of the link with r ≤ 2 r\leq 2 vanish. This is a geometric characterization for links whose Milnor invariants with r ≤ 2 r\leq 2 vanish. The chief ingredient in our proof is Habiro’s clasper theory. We also give an alternate proof of a link-homotopy classification of string links by using clasper theory.

Classification of string links up to self delta-moves and concordance

For an n–component link L, the Milnor x –invariant x L.I/ is defined for each sequence I D i1i2 im .ij 2 f1; : : : ; ng/; see Milnor [13; 14]. Let r.I/ denote the maximum number of times that any index appears. For example, r.1123/ D 2 and r.1231223/ D 3. It is known that if r.I/ D 1, then x L.I/ is a link-homotopy invariant [13], where link-homotopy is an equivalence relation on links generated by self crossing changes. Similarly, for a string link L, the Milnor –invariant L.I/ is defined; see Habegger and Lin [7]. Milnor –invariants give a link-homotopy classification for string links.

Heegaard–Floer homology and string links

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 develop similar composition formulas and long exact sequences as in knot Floer homology.

Structure of the string link concordance group and Hirzebruch-type invariants

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. We illustrate that our invariant reveals much information that Harvey's ρ n -invariant does not extract, by showing that the kernel of the ρ n -invariant is large enough to contain a subgroup with infinite rank abelianization modulo local knots. As another application, we show that concordance classes of recently discovered non-slice iterated Bing doubles are independent in an appropriate sense.

The Magnus representation and higher-order Alexander invariants for homology cobordisms of surfaces

The set of homology cobordisms from a surface to itself with markings of their boundaries has a natural monoid structure. To investigate the structure of this monoid, we define and study its Magnus representation and Reidemeister torsion invariants by generalizing Kirk-Livingston-Wang's argument over the Gassner representation of string links. Moreover, by applying Cochran and Harvey's framework of higher-order (non-commutative) Alexander invariants to them, we extract several pieces of information about the monoid and related objects.

TREE DIAGRAMS FOR STRING LINKS II: DETERMINING CHORD DIAGRAMS

In previous work [7], 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).