Grope cobordism of classical knots

By James F. Conant, P. Teichner. 2000

58 citations recorded by the source corpus.

Motivatedby the lower central series of a group, we d e/ne the notion of a grope cobordism between two knots in a 3-manifold. Just like an iterated group commutator, each grope cobordism has a type that can be d escribedby a rootedunitrivalent tree. By /ltering these trees in d i2erent ways, we show how the Goussarov– Habiro approach to /nite type invariants of knots is closely relatedto our notion of grope cobord ism. Thus our results can be viewedas a geometric interpretation of /nite type invariants. The derived commutator series of a group also has a three-dimensional analogy, namely knots modulo symmetric grope cobord ism. On one handthis theory maps onto the usual Vassiliev theory andon the other handit maps onto the Cochran–Orr–Teichner /ltration of the knot concord ance group, via symmetric grope cobordism in 4-space. In particular, the graded theory contains information on /nite type invariants (with degree h terms mapping to Vassiliev degree 2 h ), Blanch/eldforms or S-equivalence at h = 2, Casson–Gordon

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a00eb841ed9382592ec9c74eeb822a058063eca6

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Citations in the corpus

Virtual knot theory-unsolved problems (2005, unknown): “One of the more promising directions for relating Vassiliev invariants to our present concerns is the theory of gropes [14], where one considers surfaces”
Problems on invariants of knots and 3-manifolds (2004, background): “surgery along a tree clasper with d trivalent vertices whose leaves are disc-leaves [165], (2) relation on a certain collection of d crossing changes (Goussarov’s (d−1)equivalence) [150, 151, 153], (3) surgery by an element in the dth group in the lower central series of pure braid group [371], (4) ”
Noncommutative knot theory. (2002, background): “Therefore η bounds in E(K), a map of a symmetric n–stage grope [CTe].”
On Knots with trivial Alexander polynomial (2002, background): “cond stages are embedded disjointly from the Seifert surface. However, they are allowed to intersect each other. So this condition is quite diﬀerent to the notion of a “grope cobordism” introduced in [CT]. In a forthcoming paper, we will study related questions for boundary links. This is made possi”
Knot Diagrammatics (2004, background): “One of the more promising directions for relating Vassiliev invariants to our present concerns is the theory of gropes [21, 22], where one considers surfaces spanning a given knot, and then recursively the surfaces spanning curves embedded in the given surface.”
New perspectives on self-linking (2003, background): “[33] † Simple clasper surgeries of degree n. [18] † Capped grope cobordism of class n. [ 8 ]”