arXiv research

Stability theorem for concordance embeddings with applications.

We can construct a 4-manifold by attaching 2-handles to a 4-ball with framing r along the components of a link in the boundary of the 4-ball. We define a link as r-shake slice if there exists embedded spheres that represent the generators of the second homology of the 4-manifold. This naturally extends r-shake slice, a…

We define a notion of concordance based on Euler characteristic, and show that it gives rise to a concordance group of links in the three-sphere, which has the concordance group of knots as a direct summand with infinitely generated complement. We consider variants of this using oriented and nonoriented surfaces as wel…

Smoothly embed 3-manifolds in 5-manifolds, simplifying topological to smooth.

The paper classifies surfaces in 4-manifolds up to concordance.

A crucial step in the surgery-theoretic program to classify smooth manifolds is that of representing a middle--dimensional homology class by a smoothly embedded sphere. This step fails even for the simple 4-manifolds obtained from the 4-ball by adding a 2-handle with framing r along some knot K in S^3. An r-shake slice…

Study ribbon homology concordances using link Floer homology.

Fixing two concordant links in $3$--space, we study the set of all embedded concordances between them, as knotted annuli in $4$--space. When regarded up to surface-concordance or link-homotopy, the set $\mathcal{C}(L)$ of concordances from a link $L$ to itself forms a group. In order to investigate these groups, we def…

We define the concordance crosscap number of a knot as the minimum crosscap number among all the knots concordant to the knot. The four-dimensional crosscap number is the minimum first Betti number of non-orientable surfaces smoothly embedded in 4-dimensional ball, bounding the knot. Clearly the 4-dimensional crosscap …

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 ob…

The paper establishes a new pseudoisotopy result for embedding spaces, leading to computations of homotopy groups of long knots.

Study fundamental quandle of ribbon concordances, proving homomorphisms.

New method uses rational Witt span to bound concordance crosscap number of knots.

2-spheres in 4-manifolds have complete concordance obstructions if they have immersed dual spheres.

The paper confirms a conjecture about knots in aspherical 3-manifolds.

Most of the 50-year history of the study of the set of knot concordance classes, C, has focused on its structure as an abelian group. Here we take a different approach, namely we study C as a metric space admitting many natural geometric operators, especially satellite operators. We consider several knot concordance sp…

To a special type of grope embedded in 4-space, that we call an admissible grope, we associate a length function for each real number q at least 1. This gives rise to a family of pseudo-metrics d^q, refining the slice genus metric, on the set of concordance classes of knots, as the infimum of the length function taken …

We prove a concordance analogue of Gabai's $4$-dimensional light bulb theorem. That is, we show that when $R$ and $R'$ are homotopically (smoothly) embedded $2$-spheres in a $4$-manifold $X^4$ where $π_1(X^4)$ has no $2$-torsion and one of $R$ or $R'$ has a transverse sphere, then $R$ and $R'$ are concordant. When $π_1…

Many open problems and important theorems in low-dimensional topology have been formulated as statements about certain 2--complexes called gropes. This paper describes a precise correspondence between embedded gropes in 4--manifolds and the failure of the Whitney move in terms of iterated `towers' of Whitney disks. The…

New knot concordance invariants from Seiberg-Witten theory bound slice genus.

We prove a concordance version of the 4-dimensional light bulb theorem for $π_1$-negligible compact orientable surfaces, where there is a framed but not necessarily embedded dual sphere. That is, we show that if $F_0$ and $F_1$ are such surfaces in a 4-manifold $X$ that are homotopic and there exists an immersed framed…

New knots found that are 4-genus minimal.

We construct many examples of non-slice knots in 3-space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3-dimensional topological knot concordance group. The bottom part of the filtration exhibits all…

Link concordance equals homotopy for high-dimensional spheres.

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…

New surfaces in 3D manifolds are found that cannot be smoothly deformed into each other.

A cobordism between links in thickened surfaces consists of a surface $S$ and a $3$-manifold $M$, with $S$ properly embedded in $M \times I$. We show that there exist links in thickened surfaces such that if $(S,M)$ is a cobordism between them in which $S$ is simple, then $M$ must be complex. That is, there…

We propose and analyze a structure with which to organize the difference between a knot in the 3-sphere bounding a topologically embedded 2-disk in the 4-ball and it bounding a smoothly embedded disk. The n-solvable filtration of the topological knot concordance group, due to Cochran-Orr-Teichner, may be complete in th…

A geometric characterization of the Arf invariant of a knot in the 3-sphere is given in terms of two kinds of 4-dimensional bordisms, half-gropes and Whitney towers. These types of bordisms have associated complexities class and order which filter the condition of bordism by an embedded annulus, i.e. knot concordance, …

We define homotopy-theoretic invariants of knots in prime 3-manifolds. Fix a knot J in a prime 3-manifold M. Call a knot K in M concordant to J if it cobounds a properly embedded annulus with J in MxI, and call K J-characteristic if there is a degree-one map f:M --> M throwing K onto J and mapping M-K to M-J. These inv…

A knot in a thickened surface $K$ is a smooth embedding $K:S^1 \rightarrow Σ\times [0,1]$, where $Σ$ is a closed, connected, orientable surface. There is a bijective correspondence between knots in $S^2 \times [0,1]$ and knots in $S^3$, so one can view the study of knots in thickened surfaces as an extension of classic…

Computes homotopy groups of embedding spaces of arcs or circles in 4-manifolds.

New invariants lift Milnor invariants for 3-component links.

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 fre…

The concordance genus of a knot is the least genus of any knot in its concordance class. It is bounded above by the genus of the knot, and bounded below by the slice genus, two well-studied invariants. In this paper we consider the concordance genus of 11--crossing prime knots. This analysis resolves the concordance ge…

This work shows non-abelian quotient groups in string link concordance.

Khovanov homology shows (4,5) torus knot is a summand in its concordance class.

Study shows knots with similar Blanchfield forms can be homotopy ribbon concordant.

New homomorphism from Khovanov homology for knot concordance.

The concordance genus of a knot K is the minimum Seifert genus of all knots smoothly concordant to K. Concordance genus is bounded below by the 4-ball genus and above by the Seifert genus. We give a lower bound for the concordance genus of K coming from the knot Floer complex of K. As an application, we prove that ther…

Kirby and Lickorish showed that every knot in the 3-sphere is concordant to a prime knot, equivalently, every concordance class contains a prime knot. We prove here that their result can be strengthened: Every knot in the 3-sphere is invertibly concordant to a prime knot. A consequence is that every double concordance …

Abstract: Proves non-abelian group of equivariant concordance.

The concordance genus of a knot is the least genus of any knot in its concordance class. Although difficult to compute, it is a useful invariant that highlights the distinction between the three-genus and four-genus. In this paper we define and discuss the stable concordance genus of a knot, which describes the behavio…

Introduces slice knots and concordance, linking to exotic smooth structures.

New hyperbolic knots not concordant to algebraic ones found.

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-manif…

We define an infinite family of linearly independent, integer-valued smooth concordance homomorphisms. Our homomorphisms are explicitly computable and rely on local equivalence classes of knot Floer complexes over the ring $\mathbb{F}[U, V]/(UV=0)$. We compare our invariants to other concordance homomorphisms coming fr…

A fibered concordance of knots, introduced by Harer, is a concordance between fibered knots that is well-behaved with respect to the fibrations. We consider semi-fibered concordance of two component ordered links $L=J \sqcup K$ with $J$ fibered. These are concordances that restrict to fibered concordances on the first …