Knots, von Neumann signatures, and grope cobordism

Grope cobordism of classical knots

We explain the notion of a grope cobordism between two knots in a 3-manifold. Each grope cobordism has a type that can be described by a rooted unitrivalent tree. By filtering these trees in different ways, we show how the Goussarov-Habiro approach to finite type invariants of knots is closely related to our notion of grope cobordism. Thus our results can be viewed as a geometric interpretation of finite type invariants. An interesting refinement we study are knots modulo symmetric grope cobordism in 3-space. On one hand this theory maps onto the usual Vassiliev theory and on the other hand it…

Grope Cobordism and Feynman Diagrams

We explain how the usual algebras of Feynman diagrams behave under the grope degree introduced in "Grope cobordism of classical knots." We show that the Kontsevich integral rationally classifies grope cobordisms of knots in 3-space when the ``class'' is used to organize gropes. This implies that the grope cobordism equivalence relations are highly nontrivial in dimension three. We also show that the class is not a useful organizing complexity in four dimensions since only the Arf invariant survives. In contrast, measuring gropes according to ``height'' does lead to very interesting four-dimens…

Whitney towers and gropes in 4--manifolds

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 `flexibility' of these Whitney towers is used to demonstrate some geometric consequences for knot and link concordance connected to n-solvability, k-cobordism and grope concordance. The key observation is that the essential structure of gropes and Whitney towers can be described…

嵌入学与结的grope同调

Embedding calculus and grope cobordism of knots

本文证明了嵌入学不变量在长结中的满射性，并揭示了其在计算和理论上的更强能力。

Homology Cobordism Invariants and the Cochran-Orr-Teichner Filtration of the Link Concordance Group

For any group G, we define a new characteristic series related to the derived series, that we call the torsion-free derived series of G. Using this series and the Cheeger-Gromov rho-invariant, we obtain new real-valued homology cobordism invariants rho_n for closed (4k-1)-dimensional manifolds. For 3-dimensional manifolds, we show that {rho_n} is a linearly independent set and for each n, the image of rho_n is an infinitely generated and dense subset of R. In their seminal work on knot concordance, T. Cochran, K. Orr, and P. Teichner define a filtration F_(n)^m of the m-component (string) link…

Knot concordance and von Neumann $ρ$-invariants

We prove the nontriviality, at all integral levels n, of the filtration, F_n, of the classical topological knot concordance group recently defined by the authors and Kent Orr [COT]. Recall that this filtration is significant not only because of it's strong connection to Whitney tower constructions of Casson and Freedman, but also because all previously-known concordance invariants are related to the first few terms in the filtration. In [COT] we proved nontriviality at the first new level (n=3) by using von Neumann $ρ$-invariants of the 3-manifolds obtained by zero surgery on the knots. Here, …