Grope cobordism of classical knots

Abstract

Motivated by the lower central series of a group, we define 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 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.The derived commutator series of a group also has a 3-dimensional analogy, namely knots modulo symmetric grope cobordism. On one hand this theory maps onto the usual Vassiliev theory and on the other hand it maps onto the Cochran-Orr-Teichner filtration of the knot concordance group, via symmetric grope cobordism in 4-space. In particular, the graded theory contains information on finite type invariants (with degree h terms mapping to Vassiliev degree 2 h ), Blanchfield forms or S-equivalence at h = 2, Casson-Gordon invariants at h = 3, and for h = 4 one finds the new von Neumann signatures of a knot.1991 Mathematics Subject Classification. 57M27.

Simple Whitney towers, half-gropes and the Arf invariant of a knot

“…The equivalence of (i) and (ii) also follows indirectly from results in [Conant and Teichner 2004b;2004a;Cochran et al 2003]; see particularly [Conant and Teichner 2004a, Proposition 3.8].…”

Whitney towers and gropes in 4–manifolds

“…In light of these notational conventions, it turns out to be convenient to associate to each grope a disjoint union of unitrivalent trees (as in [6], [7] and [8]) rather than the customary (single) multivalent tree (e.g. in [15]).…”

“…A dyadic (capped) A-like grope g is a half-grope if all the trees in t(g) are simple (right-or left-normed) as illustrated in Figure 17; note that the roots (shown pointing down in the figure) are required to be at an "end" of the tree. The proof of Proposition 7.1 uses the geometric IHX Lemma 7.2 below to follow the algebraic proof that the usual group of unitrivalent trees occurring in finite type theory is spanned by simple trees as given in e.g., [1], [7]:…”

Gropes and the rational lift of the Kontsevich integral

“…Suppose a knot bounds an embedded grope of class 2n in S 3 , where all the surface stages are of genus one. Then the knot can be represented as a rooted tree clasper surgery, T , in the complement of the unknot which forms a meridian to the root leaf [2]. Note that the other leaves can be embedded arbitrarily in the complement of the unknot's spanning disk.…”

“…We conclude by emphasizing the fact that this note deals with knots that bound gropes, which is a more restricted class than those cobounding a grope with the unknot, as studied in [2].…”