Grope cobordism and feynman diagrams

Conant, James; Teichner, Peter

doi:10.1007/s00208-003-0477-y

Cited by 22 publications

(50 citation statements)

References 16 publications

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“…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].…”

Section: Introductionmentioning

confidence: 64%

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

Schneiderman¹

2005

Pacific J. Math.

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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, and it is shown constructively that the Arf invariant is exactly the obstruction to cobording pairs of knots by half-gropes and Whitney towers of arbitrarily high class and order, respectively. This illustrates geometrically how, in the setting of knot concordance, the Vassiliev (isotopy) invariants "collapse" to the Arf invariant.

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Section: Introductionmentioning

confidence: 64%

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

Schneiderman¹

2005

Pacific J. Math.

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“…In this section we will review definitions of twisted and framed asymmetric Whitney towers in 4-manifolds, and discuss uni-trivalent trees which arise naturally in the study of iterated intersections of surfaces, particularly for Whitney towers (e.g., see [Coc90,CT04a,CT04b,Sch06,CST07,CST12c,CST14]). Readers who are familiar with them may skip to Section 3, after reading this paragraph.…”

Section: Whitney Towers and Associated Treesmentioning

confidence: 99%

Rational Whitney tower filtration of links

Choon

2017

Math. Ann.

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Abstract. We present complete classifications of links in the 3-sphere modulo framed and twisted Whitney towers in a rational homology 4-ball. This provides a geometric characterization of the vanishing of the Milnor invariants of links in terms of Whitney towers. Our result also says that the higher order Arf invariants, which are conjectured to be nontrivial, measure the potential difference between the Whitney tower theory in rational homology 4-balls and that in the 4-ball extensively developed by Conant, Schneiderman and Teichner. IntroductionTopology of dimension 4 is different from high dimensions because the Whitney move may fail. The essential problem is to find an embedded Whitney disk along which a pair of intersections of two sheets could be removed by a Whitney move. Once an immersed Whitney disk is obtained from fundamental group data, one may try to remove double points of the disk by finding a next stage of Whitney disks. Iterating this, we are led to the notion of a Whitney tower.Since work of Cochran, Orr and Teichner [COT03], concordance of knots and links, which is the "local case" of general disk embedding, has been extensively studied via frameworks formulated in terms of Whitney towers. In this paper, we will focus on asymmetric Whitney towers in dimension 4 bounded by links in S 3 , motivated from work of Conant, Schneiderman and Teichner [CST11,CST12c,CST14,CST12b,CST12a]. Whitney towers come in two flavors: framed and twisted. Whitney towers we consider have an order, which is a nonnegative integer measuring the number of iterated stages. Precise definitions can be found in Section 2.The main result of this paper is a complete classification of links in S 3 modulo Whitney towers in rational homology 4-balls. To state our result, we use the following notation. Fix m > 0, and let W n be the set of m-component links in S 3 bounding a twisted Whitney tower of order n in a rational homology 4-ball with boundary S 3 . We define the graded quotient W n of W n by the condition that L and L ′ in W n represent the same element in W n if and only if a band sum of L and −L ′ lies in W n+1 . In fact, in Section 4.3, we will show that it is an equivalence relation, and L ∈ W n+1 if and only if [L] = 0 in W n . So we may write W n = W n /W n+1 . Theorem A.(1) Band sum is a well-defined operation on the set W n , independent of the choice of bands, and W n is an abelian group under band sum. (2) W n is classified by the Milnor invariants of order n (= length n + 2). (3) W n is a free abelian group of rank mR(m, n + 1) − R(m, n + 2), where R(m, n) = We remark that R(m, n) is the rank of the degree n part of the free Lie algebra on m variables, due to Witt (e.g., see [MKS66, Section 5.6]), and mR(m, n + 1) − R(m, n + 2)2010 Mathematics Subject Classification. 57N13, 57N70, 57M25. Key words and phrases. Whitney towers, Concordance, Rational homology 4-balls, Links, Milnor invariants, higher order Arf invariants. RATIONAL WHITNEY TOWER FILTRATION OF LINKS2 is the number of linearly independent Milnor invar...

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“…Proof. By geometric IHX relations [2,8] (which preserve framings) one can write M C = M C 1 ∪···∪C k , where each C i is a clasper of order n with a +1-framed unknotted leaf adjacent to a trivalent vertex which is adjacent to another leaf. The +1-framed unknotted leaf of each C i bounds an embedded disk which intersects the other claspers and may even have interior intersections with C i .…”

Section: Lemma 19mentioning

confidence: 99%

Geometric filtrations of string links and homology cylinders

2016

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ABSTRACT. We show that the Artin representation on concordance classes of string links induces a well-defined epimorphism modulo order n twisted Whitney tower concordance, and that the kernel of this map is generated by band sums of iterated Bing-doubles of any string knot with nonzero Arf invariant. We also continue J. Levine's work [20,21,22] comparing two filtrations of the group of homology cobordism classes of 3-dimensional homology cylinders, one defined in terms of an Artin-type representation (the Johnson filtration) and one defined using clasper surgery (the Goussarov-Habiro filtration). In particular, the associated graded groups are completely classified up to an unknown 2-torsion summand for the Goussarov-Habiro filtration, for which we obtain an upper bound, in a precisely analogous fashion to the classification of the Whitney tower filtration of link concordance.

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Grope cobordism and feynman diagrams

Cited by 22 publications

References 16 publications

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

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

Rational Whitney tower filtration of links

Geometric filtrations of string links and homology cylinders

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