We consider locally linear Z_p x Z_p actions on the four-sphere. We present simple constructions of interesting examples, and then prove that a given action is concordant to its linear model if and only if a single surgery obstruction taking to form of an Arf invariant vanishes. We discuss the behavior of this invariant under various connected-sum operations, and conclude with a brief discussion of the existence of actions which are not concordant to their linear models.
Any knot in a solid torus, called a pattern or satellite operator, acts on knots in the 3-sphere via the satellite construction. We introduce a generalization of satellite operators which form a group (unlike traditional satellite operators), modulo a generalization of concordance. This group has an action on the set of knots in homology spheres, using which we recover the recent result of Cochran and the authors that satellite operators with strong winding number $\pm 1$ give injective functions on topological concordance classes of knots, as well as smooth concordance classes of knots modulo…
We describe an action of the concordance group of knots in the three-sphere on concordances of knots in arbitrary 3-manifolds. As an application we define the notion of almost-concordance between knots. After some basic results, we prove the existence of non-trivial almost-concordance classes in all non-abelian 3-manifolds. Afterwards, we focus the attention on the case of lens spaces, and use a modified version of the Ozsvath-Szabo-Rasmussen's tau-invariant to obstruct almost-concordances and prove that each L(p,1) admits infinitely many nullhomologous non almost-concordant knots. Finally we …
We investigate the disparity between smooth and topological almost concordance of knots in general 3-manifolds Y. Almost concordance is defined by considering knots in Y modulo concordance in Yx[0,1] and the action of the concordance group of knots in the 3-sphere that ties in local knots. We prove that the trivial free homotopy class in every 3-manifold other than the 3-sphere contains an infinite family of knots, all topologically concordant, but not smoothly almost concordant to one another. Then, in every lens space and for every free homotopy class, we find a pair of topologically concord…
In this note, we combine the recent 4-dimensional light bulb theorem of David Gabai and a recent construction of concordances for knots in $S^2\times S^1$ due to Eylem Zeliha Yildiz to construct a concordance between the standard surface of genus $g$ in $S^2\times S^2$ and any homologous surface.
Knots and links in 3-manifolds are studied by applying intersection invariants to singular concordances. The resulting link invariants generalize the Arf invariant, the mod 2 Sato-Levine invariants, and Milnor's triple linking numbers. Besides fitting into a general theory of Whitney towers, these invariants provide obstructions to the existence of a singular concordance which can be homotoped to an embedding after stabilization by connected sums with $S^2\times S^2$. Results include classifications of stably slice links in orientable 3-manifolds, stable knot concordance in products of an orie…
Let $Σ_0,Σ_1$ be closed oriented surfaces. Two oriented knots $K_0 \subset Σ_0 \times [0,1]$ and $K_1 \subset Σ_1 \times [0,1]$ are said to be (virtually) concordant if there is a compact oriented $3$-manifold $W$ and a smoothly and properly embedded annulus $A$ in $W \times [0,1]$ such that $\partial W=Σ_1 \sqcup -Σ_0$ and $\partial A=K_1 \sqcup -K_0$. This notion of concordance, due to Turaev, is equivalent to concordance of virtual knots, due to Kauffman. A prime virtual knot, in the sense of Matveev, is one for which no thickened surface representative $K \subset Σ\times [0,1]$ admits a no…
We study Khovanov homology over the Frobenius algebra $\mathbb{F}[U,V,X]/((X-U)(X-V))$, or $U(1) \times U(1)$-equivariant Khovanov homology, and extract two families of concordance invariants using the algebraic $U$-power and $V$-power filtrations on the chain complex. We also further develop the reduced version of the theory and study its behavior under mirroring.
Using the Gordon-Litherland pairing, one can define invariants (signature, nullity, determinant) for ${\mathbb Z}/2$ null-homologous links in thickened surfaces. In this paper, we study the concordance properties of these invariants. For example, if $K \subset Σ\times I$ is ${\mathbb Z}/2$ null-homologous and slice, we show that its signatures vanish and its determinants are perfect squares. These statements are derived from a cobordism result for closed unoriented surfaces in certain 4-manifolds. The Brown invariants are defined for ${\mathbb Z}/2$ null-homologous links in thickened surfaces.…
We present a new property, the Disjoint Path Concordances Property, of an ENR homology manifold X which precisely characterizes when X times R has the Disjoint Disks Property. As a consequence, X times R is a manifold if and only if X is resolvable and it possesses this Disjoint Path Concordances Property.
Milnor's $\barμ$-invariants of links in the $3$-sphere $S^3$ vanish on any link concordant to a boundary link. In particular, they are trivial on any knot in $S^3$. Here we consider knots in thickened surfaces $Σ\times [0,1]$, where $Σ$ is closed and oriented. We construct new concordance invariants by adapting the Chen-Milnor theory of links in $S^3$ to an extension of the group of a virtual knot. A key ingredient is the Bar-Natan $\textit{Zh}$ map, which allows for a geometric interpretation of the group extension. The group extension itself was originally defined by Silver-Williams. Our ext…
It is known that there is a unique concordance class in the free homotopy class of $S^1\times pt \subset S^1 \times S^2$. The constructive proof of this fact is given by the second author. It turns out that all the concordances in this construction are invertible. The knots $K\subset S^{1}\times S^{2}$ with hyperbolic complements and trivial symmetry group are special interest here, because they can be used to generate absolutely exotic compact 4-manifolds by the recipe given by Akbulut and Ruberman. Here we built absolutely exotic manifold pairs by this construction, and show that this constr…
The knot Floer complex and the concordance invariant $\varepsilon$ can be used to define a filtration on the smooth concordance group. We exhibit an ordered subset of this filtration that is isomorphic to $\mathbb{N} \times \mathbb{N}$ and consists of topologically slice knots.
Let $K$ be a knot in an integral homology 3-sphere $Y$, and $Σ$ the corresponding $n$-fold cyclic branched cover. Assuming that $Σ$ is a rational homology sphere (which is always the case when $n$ is a prime power), we give a formula for the Lefschetz number of the action that the covering translation induces on the reduced monopole homology of $Σ$. The proof relies on a careful analysis of the Seiberg--Witten equations on 3-orbifolds and of various $η$-invariants. We give several applications of our formula: (1) we calculate the Seiberg--Witten and Furuta--Ohta invariants for the mapping tori…
