Link Cobordism Research Articles

Towards an [formula omitted] action on the annular Khovanov spectrum

Given a link in the thickened annulus, its annular Khovanov homology carries an action of the Lie algebra sl2, which is natural with respect to annular link cobordisms. We consider the problem of lifting this action to the stable homotopy refinement of the annular homology. As part of this program, the actions of the standard generators of sl2 are lifted to maps of spectra. In particular, it follows that the sl2 action on homology commutes with the action of the Steenrod algebra. The main new technical ingredients developed in this paper, which may be of independent interest, concern certain types of cancellations in the cube of resolutions and the resulting more intricate structure of the moduli spaces in the framed flow category.

Classification of virtual string links up to cobordism

Cobordism of virtual string links on n strands is a combinatorial generalization of link cobordism. There exists a bijection between virtual string links on n strands up to cobordisms and elements of the direct product of n ( n − 1) copies of the integers. 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.

Contact handles, duality, and sutured Floer homology

We give an explicit construction of the Honda--Kazez--Mati\'c gluing maps in terms of contact handles. We use this to prove a duality result for turning a sutured manifold cobordism around, and to compute the trace in the sutured Floer TQFT. We also show that the decorated link cobordism maps on the hat version of link Floer homology defined by the first author via sutured manifold cobordisms and by the second author via elementary cobordisms agree.

Multivariable Signatures, Genus Bounds, and 0.5-Solvable Cobordisms

We refine prior bounds on how the multivariable signature and the nullity of a link change under link cobordisms. The formula generalizes a series of results about the 4-genus having their origins in the Murasugi–Tristram inequality, and at the same time extends previously known results about concordance invariance of the signature to a bigger set of allowed variables. Finally, we show that the multivariable signature and nullity are also invariant under 0.5-solvable cobordism.

Link cobordisms and absolute gradings on link Floer homology

We show that the link cobordism maps defined by the author are graded and satisfy a grading change formula. Using the grading change formula, we prove a new bound for \Upsilon_K(t) for knot cobordisms in negative definite 4-manifolds. As another application, we show that the link cobordism maps associated to a connected, closed surface in S^4 are determined by the genus of the surface. We also prove a new adjunction relation and adjunction inequality for the link cobordism maps. Along the way, we see how many known results in Heegaard Floer homology can be proven using basic properties of the link cobordism maps, together with the grading change formula.

On the functoriality of Khovanov–Floer theories

We introduce the notion of a Khovanov–Floer theory. We prove that every page (after E1) of the spectral sequence accompanying a Khovanov–Floer theory is a link invariant, and that an oriented link cobordism induces a map on each page which is an invariant of the cobordism up to smooth isotopy rel boundary. We then prove that the spectral sequences relating Khovanov homology to Heegaard Floer homology and singular instanton knot homology are induced by Khovanov–Floer theories and are therefore functorial in the manner described above, as had been conjectured for some time.

Link cobordisms and functoriality in link Floer homology

We construct cobordism maps on link Floer homology associated to decorated link cobordisms. The maps are defined on a curved chain homotopy type invariant. We describe the construction and prove invariance. We also make a comparison with the graph TQFT for Heegaard Floer homology.

Computing cobordism maps in link Floer homology and the reduced Khovanov TQFT

We study the maps induced on link Floer homology by elementary decorated link cobordisms. We compute these for births, deaths, stabilizations, and destabilizations, and show that saddle cobordisms can be computed in terms of maps in a decorated skein exact triangle that extends the oriented skein exact triangle in knot Floer homology. In particular, we completely determine the Alexander and Maslov grading shifts. As a corollary, we compute the maps induced by elementary cobordisms between unlinks. We show that these give rise to a $$(1+1)$$ -dimensional TQFT that coincides with the reduced Khovanov TQFT. Hence, when applied to the cube of resolutions of a marked link diagram, it gives the complex defining the reduced Khovanov homology of the knot. Finally, we define a spectral sequence from (reduced) Khovanov homology using these cobordism maps, and we prove that it is an invariant of the (marked) link.

Cobordisms of sutured manifolds and the functoriality of link Floer homology

It has been a central open problem in Heegaard Floer theory whether cobordisms of links induce homomorphisms on the associated link Floer homology groups. We provide an affirmative answer by introducing a natural notion of cobordism between sutured manifolds, and showing that such a cobordism induces a map on sutured Floer homology. This map is a common generalization of the hat version of the closed 3-manifold cobordism map in Heegaard Floer theory, and the contact gluing map defined by Honda, Kazez, and Matić. We show that sutured Floer homology, together with the above cobordism maps, forms a type of TQFT in the sense of Atiyah. Applied to the sutured manifold cobordism complementary to a decorated link cobordism, our theory gives rise to the desired map on link Floer homology. Hence, link Floer homology is a categorification of the multi-variable Alexander polynomial. We outline an alternative definition of the contact gluing map using only the contact element and handle maps. Finally, we show that a Weinstein sutured manifold cobordism preserves the contact element.

Movie moves for singular link cobordisms in 4-dimensional space

Two singular links are cobordant if one can be obtained from the other by singular link isotopy together with a combination of births or deaths of simple unknotted curves, and saddle point transformations. A movie description of a singular link cobordism in 4-space is a sequence of singular link diagrams obtained from a projection of the cobordism into 3-space by taking 2-dimensional cross-sections perpendicular to a fixed direction. We present a set of movie moves that are sufficient to connect any two movies of isotopic singular link cobordisms.

Braiding link cobordisms and non-ribbon surfaces

We define the notion of a braided link cobordism in $S^3 \times [0,1]$, which generalizes Viro's closed surface braids in $\mathbb{R}^4$. We prove that any properly embedded oriented surface $W \subset S^3 \times [0,1]$ is isotopic to a surface in this special position, and that the isotopy can be taken rel boundary when $\partial W$ already consists of closed braids. These surfaces are closely related to another notion of surface braiding in $D^2 \times D^2$, called braided surfaces with caps, which are a generalization of Rudolph's braided surfaces. We mention several applications of braided surfaces with caps, including using them to apply algebraic techniques from braid groups to studying surfaces in 4-space, as well as constructing singular fibrations on smooth 4-manifolds from a given handle decomposition.

Categorification of the Colored Jones Polynomial and Rasmussen Invariant of Links

AbstractWe define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphismclasses of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further, we study conditions under which framed colored link cobordisms induce chain transformations between our formal brackets. We conjecture that for special choice of parameters, Khovanov and Lee homology theories of colored links are functorial (up to sign). Finally, we extend the Rasmussen invariant to links and give examples where this invariant is a stronger obstruction to sliceness than the multivariable Levine–Tristram signature.

The foam and the matrix factorizationsl3link homologies are equivalent

We prove that the foam and matrix factorization universal rational sl3 link homologies are naturally isomorphic as projective functors from the category of link and link cobordisms to the category of bigraded vector spaces.

Khovanov-Jacobsson numbers and invariants of surface-knots derived from Bar-Natan’s theory

Khovanov introduced a cohomology theory for oriented classical links whose graded Euler characteristic is the Jones polynomial. Since Khovanov’s theory is functorial for link cobordisms between classical links, we obtain an invariant of a surface-knot, called the Khovanov-Jacobsson number, by considering the surface-knot as a link cobordism between empty links. In this paper, we study an extension of the Khovanov-Jacobsson number derived from Bar-Natan’s theory, and prove that any T 2 T^2 -knot has trivial Khovanov-Jacobsson number.

An invariant of link cobordisms from Khovanov homology

In [Duke Math. J. 101 (1999) 359-426], Mikhail Khovanov constructed a homology theory for oriented links, whose graded Euler characteristic is the Jones polynomial. He also explained how every link cobordism between two links induces a homomorphism between their homology groups, and he conjectured the invariance (up to sign) of this homomorphism under ambient isotopy of the link cobordism. In this paper we prove this conjecture, after having made a necessary improvement on its statement. We also introduce polynomial Lefschetz numbers of cobordisms from a link to itself such that the Lefschetz polynomial of the trivial cobordism is the Jones polynomial. These polynomials can be computed on the chain level.

Invariants of boundary link cobordism

Introduction Main results Preliminaries Morita Equivalence Devissage Varieties of representations Generalizing Pfister's theorem Characters Detecting rationality and integrality Representation varieties: Two examples Number theory invariants All division algebras occur Appendix I. Primitive element theorems Appendix II. Hermitian categories Bibliography Index.

Link Cobordism and the Intersection of Slice Discs

We work in the smooth category. An (oriented) (ordered) m-component n-(dimensional) link is a smooth oriented submanifold L = {K1, …, Km} of Sn+2 which is the ordered disjoint union of m manifolds, each PL-homeomorphic to the standard n-sphere. If m = 1, then L is called a knot. We say that m-component n-dimensional links L0 and L1 are (link-)concordant or (link-)cobordant if there is a smooth oriented submanifold C = {C1, …, Cm} of Sn+2 × [0, 1] which meets the boundary transversely in ∂C, is PL-homeomorphic to L0 × [0, 1], and meets Sn+2 × {l} in Ll (l = 0, 1). If m = 1, then we say that n-knots L0 and Ll are (knot-)concordant or (knot-)cobordant. Then we call C a concordance-cylinder of the two n-knots L0 and Ll. If an n-link L is concordant to the trivial link, then we call L a slice link. If an n-link L = {K1, …, Km} ⊂ Sn+2 = ∂Bn+3 ⊂ Bn+3 is slice, then there is a disjoint union of (n + 1)-discs D 1 n + 1 II ··· II D m n + 1 in Bn+3 such that D 1 n + 1 ∩ S n + 2 = ∂ D i n + 1 = K i . ( D 1 n + 1 , … , D m n + 1 ) is called a set of slice discs for L. If m = 1, then D 1 n + 1 is called a slice disc for the knot L. 1991 Mathematics Subject Classification 57M25, 57Q45.

Rokhlin's inequalities for real algebraic curves and¶an extremal property

Rokhlin's inequalities for real algebraic curves and an extremal property for one of the inequalities are derived via a link cobordism approach.

Représentations du groupe des tresses généralisées dans des groupes de Lie

Let Γ be a Lie group.Then any automorphism of the free group of rank n induces a diffeomorphism of Γ n . We use this remark and a result of P. Vogel to construct linear representations of a certain automorphism group of the algebraic closure of the free group. This automorphism group is closely related to the string link cobordism group.

Homology boundary links and Blanchfield forms: concordance classification and new tangle-theoretic constructions

THIS work forms part of our on-going effort to classify the set of concordance classes of links.RecallthatalinkL= (K,,. . . ,K,}inS “+’ is a locally flat piecewise-linear, oriented submanifold of S”+2 of which each component Ki is homeomorphic to S”. The exterior E(L) of a link L is the closure of the complement of a small regular neighborhood N(L) of L. A longitude of a component Ki is a parallel of Ki lying on the boundary of the tubular neighborhood (untwisted if n = 1). A meridian pi is a path from a basepoint to JN(L) which traverses a fiber of 8N(L) and returns. A Seifert Surface for Ki is a connected, compact, oriented, (n + 1)-manifold K E E(L) such that aK is a longitude of Ki. Links Lo, L1 are concordant (or cobordant) if there is a smooth, oriented submanifold C = {C,, . . . , C,} of S n+2 x [0, l] which meets the boundary transversely in X, is piecewise-linearly homeomorphic to L,, x [0, 11, and meets Sn+2 x {i} in Li for i = 0, 1. In the mid-603 M. Kervaire and J. Levine gave an algebraic classification of knot concordance groups (m = 1) in high dimensions (n > 1) [l]. For even n these are trivial and for odd n they are infinitely generated, being isomorphic to certain Witt groups obtained from information garnered from the Seifert surface. Extending Levine’s knot cobordism classification to links is difficult for several reasons. Firstly, if m > 1, the natural operation of connected-sum is not well-defined on concordance classes so there is no obvious group structure. Secondly, the Seifert surfaces for different components of a link may intersect, obstructing at least the naive generalization of the Seifert form information. However, the techniques do extend well to the class of boundary links. A boundary link is one which admits a collection of m disjoint Seifert surfaces, one for each component. In fact, S. Cappell and J. Shaneson classified boundary links modulo boundary link cobordism in 1980 using their homology surgery groups, followed later by Ki Ko and W. Mio who accomplished this via Seifert surfaces [2-41. A boundary link cobordism is a cobordism C between Lo and L1 for which there exist disjointly embedded 2n-manifolds IV= {IV,, . . . , ZVm) in E(C) such that IVn(S”+2 x {i}) is a system of Seifert surfaces for the boundary link Li, i = 0, 1, and such that