Gropes and the rational lift of the Kontsevich integral

Conant, James

doi:10.4064/fm184-0-5

Cited by 2 publications

(2 citation statements)

References 3 publications

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“…This allows for analogues of all of the above results. These other equivalence relations on knots and links have been much studied recently in many different contexts and are related to notions of gropes [33] [11]. In particular, as applications of our main theorem we show that a family of Cheeger-Gromov von Neumann ρ-invariants of links and 3-manifolds considered by Harvey in [19] are actually invariants of weaker equivalence relations involving gropes similar but more general than those considered in [7] [9] (generalizing [19,Section 6]).…”

Section: Introductionmentioning

confidence: 71%

Homology and derived series of groups II: Dwyer’s Theorem

Cochran

Harvey

2008

Geom. Topol.

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We give new information about the relationship between the low-dimensional homology of a space and the derived series of its fundamental group. Applications are given to detecting when a set of elements of a group generates a subgroup "large enough" to map onto a nonabelian free solvable group, and to concordance and grope cobordism of classical links. We also greatly generalize several key homological results employed in recent work of Cochran-Orr-Teichner in the context of classical knot concordance. .n/ H , associated to the derived series, called the torsion-free derived series. The authors previously established a precise analogue, for the torsion-free derived series, of Stallings' theorem. Here our main result is the analogue of Dwyer's theorem for the torsion-free derived series. We also prove a version of Dwyer's theorem for the rational lower central series. We apply these to give new results on the Cochran-Orr-Teichner filtration of the classical link concordance group. 57M07; 20J06, 55P60

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Section: Introductionmentioning

confidence: 71%

Homology and derived series of groups II: Dwyer’s Theorem

Cochran

Harvey

2008

Geom. Topol.

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“…Such examples can be constructed by surgery along graph claspers with at least 3 trivalent vertices; for graph claspers, see Section 4.3. More generally, it is known [7] that we can construct knots which have the same .< n/-loop part of the Kontsevich invariant, but have different n-loop part, for any n. For example, the following knot and the trivial knot give such examples, when this graph clasper has 2n trivalent vertices.…”

Section: Theorem 44 (See a Problem Inmentioning

confidence: 99%

On the 2–loop polynomial of knots

Ohtsuki

2007

Geom. Topol.

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The 2-loop polynomial of a knot is a polynomial characterizing the 2-loop part of the Kontsevich invariant of the knot. An aim of this paper is to give a methodology to calculate the 2-loop polynomial. We introduce Gaussian diagrams to calculate the rational version of the Aarhus integral explicitly, which constructs the 2-loop polynomial, and we develop methodology of calculating Gaussian diagrams showing many basic formulas of them. As a consequence, we obtain an explicit presentation of the 2-loop polynomial for knots of genus 1 in terms of derivatives of the Jones polynomial of the knots.Corresponding to quantum and related invariants of 3-manifolds, we can formulate equivariant invariants of the infinite cyclic covers of knots complements. Among such equivariant invariants, we can regard the 2-loop polynomial of a knot as an "equivariant Casson invariant" of the infinite cyclic cover of the knot complement. As an aspect of an equivariant Casson invariant, we show that the 2-loop polynomial of a knot is presented by using finite type invariants of degree Ä 3 of a spine of a Seifert surface of the knot. By calculating this presentation concretely, we show that the degree of the 2-loop polynomial of a knot is bounded by twice the genus of the knot. This estimate of genus is effective, in particular, for knots with trivial Alexander polynomial, such as the Kinoshita-Terasaka knot and the Conway knot. 57M27; 57M25Dedicated to Professor Yukio Matsumoto on the occasion of his 60th birthday IntroductionThe Kontsevich invariant is a very strong invariant of knots, which dominates all quantum invariants and all Vassiliev invariants, and it is expected that the Kontsevich invariant classifies knots. A problem when we study the Kontsevich invariant is that it is difficult to calculate the Kontsevich invariant for any knot concretely. That is, the value of the Kontsevich invariant is presented by an infinite linear sum of Jacobi diagrams (a certain kind of uni-trivalent graphs), and it is not known so far how to calculate all terms of such a linear sum at the same time for an arbitrarily given knot. Geometry & Topology, Volume 11 (2007) On the 2-loop polynomial of knots 1359by the equivariant linking number (Kojima and Yamasaki [17]) and equivariant finite type invariants of degree Ä 3, while a surgery formula for the Alexander polynomial is given by the equivariant linking number. Another aspect of an equivariant Casson invariant is that the 2-loop polynomial of a knot is presented by using finite type invariants of degree Ä 3 of a spine of a Seifert surface of the knot (Theorem 4.4), while the Alexander polynomial of a knot is presented by using finite type invariants of degree 1, ie, the Seifert form, of a spine of a Seifert surface of the knot.By constructing the 2-loop polynomial using Gaussian diagrams along the latter aspect, in Theorem 4.7, we show the following estimate, which was conjectured by Rozansky [40], that the degree of the 2-loop polynomial of a knot Ä 2 the genus of the knot ;where the genus of a kno...

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Gropes and the rational lift of the Kontsevich integral

Cited by 2 publications

References 3 publications

Homology and derived series of groups II: Dwyer’s Theorem

Homology and derived series of groups II: Dwyer’s Theorem

On the 2–loop polynomial of knots

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