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

Schneiderman, Rob

doi:10.2140/pjm.2005.222.169

Cited by 14 publications

(26 citation statements)

References 23 publications

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“…In the Whitney tower obstruction theory of [49], the order n intersection invariant τ n (W) ∈ T n assigned to each order n (framed) Whitney tower W is defined by summing the trees associated to unpaired intersections in W (see Figure 2 for an example). The tree orientations are induced by Whitney disk orientations via a convention that corresponds to the AS relations (Section 2.5), and the IHX relations can be realized geometrically by controlled maneuvers on Whitney towers as described in [10,47]. It follows from the obstruction theory that a link bounds an order n Whitney tower W with τ n (W) = 0 if and only if it bounds an order n + 1 Whitney tower [49,Thm.2].…”

Section: 2mentioning

confidence: 99%

Milnor invariants and twisted Whitney towers

Conant

Schneiderman

Teichner

2013

Journal of Topology

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This paper describes the relationship between the first non‐vanishing Milnor invariants of a classical link and the intersection invariant of a twisted Whitney tower. This is a certain 2‐complex in the 4‐ball, built from immersed disks bounded by the given link in the 3‐sphere together with finitely many ‘layers’ of Whitney disks. The intersection invariant is a higher‐order generalization of the intersection number between two immersed disks in the 4‐ball, well known to give the linking number of the link on the boundary, which measures intersections among the Whitney disks and the disks bounding the given link, together with information that measures the twists (framing obstructions) of the Whitney disks. This interpretation of Milnor invariants as higher‐order intersection invariants plays a key role in our classifications [J. Conant, R. Schneiderman and P. Teichner, ‘Higher‐order intersections in low‐dimensional topology’, Proc. Natl Acad. Sci. USA 108 (2011) 8131–8138; J. Conant, R. Schneiderman and P. Teichner, ‘Whitney tower concordance of classical links’, Geom. Topol. 16 (2012) 1419–1479] of both the framed and twisted Whitney tower filtrations on link concordance. Here, we show how to realize the higher‐order Arf invariants, which also play a role in the classifications, and derive new geometric characterizations of links with vanishing length at most 2k Milnor invariants.

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Section: 2mentioning

confidence: 99%

Milnor invariants and twisted Whitney towers

Conant

Schneiderman

Teichner

2013

Journal of Topology

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“…Alternatively, one could first exchange all twisted Whitney disks of order greater than n/2 for unpaired intersections of order greater than n by boundary-twisting ( Figure 18). Then, all intersections of order greater than n can be converted into into many algebraically canceling pairs of order n intersections by repeatedly "pushing down" unpaired intersections until they reach the order zero disks, as illustrated for instance in Figure 12 of [35] (assuming, as we may, that W contains no Whitney disks of order greater than n).…”

Section: Notation and Conventionsmentioning

confidence: 99%

Whitney tower concordance of classical links

2012

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This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new quadratic refinement of the intersection theory is introduced, measuring Whitney disk framing obstructions. It is shown that the filtrations are completely classified by Milnor invariants together with new higher-order Sato-Levine and higher-order Arf invariants, which are obstructions to framing a twisted Whitney tower in the 4-ball bounded by a link in the 3-sphere. Applications include computation of the grope filtration and new geometric characterizations of Milnor's link invariants. 57M25, 57M27, 57Q60; 57N10

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“…The reader is referred to [11] for details on immersed surfaces in 4-manifolds, including Whitney moves and (Casson) finger moves. For more on Whitney towers see [9], [28], [29].…”

Section: Whitney Towersmentioning

confidence: 99%

Whitney towers and the Kontsevich integral

Schneiderman

Teichner

2004

Geometry and Topology Monographs

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We continue to develop an obstruction theory for embedding 2-spheres into 4-manifolds in terms of Whitney towers. The proposed intersection invariants take values in certain graded abelian groups generated by labelled trivalent trees, and with relations well known from the 3-dimensional theory of finite type invariants. Surprisingly, the same exact relations arise in 4 dimensions, for example the Jacobi (or IHX) relation comes in our context from the freedom of choosing Whitney arcs. We use the finite type theory to show that our invariants agree with the (leading term of the tree part of the) Kontsevich integral in the case where the 4manifold is obtained from the 4-ball by attaching handles along a link in the 3-sphere.

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

Cited by 14 publications

References 23 publications

Milnor invariants and twisted Whitney towers

Milnor invariants and twisted Whitney towers

Whitney tower concordance of classical links

Whitney towers and the Kontsevich integral

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