Kosanović, Danica: A geometric approach to the embedding calculus knot invariants. - Bonn, 2020. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.

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@phdthesis{handle:20.500.11811/8651,

urn: https://nbn-resolving.org/urn:nbn:de:hbz:5-59798,

author = {{Danica Kosanović}},

title = {A geometric approach to the embedding calculus knot invariants},

school = {Rheinische Friedrich-Wilhelms-Universität Bonn},

year = 2020,

month = oct,

note = {In this thesis we consider two homotopy theoretic approaches to the study of spaces of knots: the theory of finite type invariants of Vassiliev and the embedding calculus of Goodwillie and Weiss, and address connections between them.

Our results confirm that the knot invariants ev

There are two crucial ingredients for this result. Firstly, we study the so-called Taylor tower of the embedding calculus more generally for long knots in any manifold with dim(M) ≥ 3 and develop a geometric understanding of its layers (fibres between two consecutive spaces in the tower). In particular, we describe their first non-vanishing homotopy groups in terms of groups of decorated trees.

Secondly, we give an explicit interpretation of ev

Our main theorem then states that the first possibly non-vanishing embedding calculus invariant of a knot which is grope cobordant to the unknot is precisely the equivalence class of the underlying decorated tree of the grope in the homotopy group of the layer. The surjectvity of ev

As another corollary we obtain a sufficient condition for the mentioned conjecture to hold over a certain coefficient group A. Namely, it is enough that the spectral sequence for the homotopy groups of the Taylor tower, tensored with A, collapses along the diagonal. In particular, such a collapse result is known for A = Q, confirming that the embedding calculus invariants are universal rational additive Vassiliev invariants, and that they factor configuration space integrals through the Taylor tower. It also follows that they are universal over the p-adic integers in a range depending on the prime p, using recent results of Boavida de Brito and Horel.

Moreover, the surjectivity of ev

Finally, we also discuss the geometric approach to the finite type theory in terms of the Gusarov–Habiro filtration of the set of isotopy classes of knots in a 3-manifold. We extend some known techniques to prove that the associated graded quotients of this filtration are abelian groups, and study the map which relates these groups to certain graph complexes.},

url = {http://hdl.handle.net/20.500.11811/8651}

}

urn: https://nbn-resolving.org/urn:nbn:de:hbz:5-59798,

author = {{Danica Kosanović}},

title = {A geometric approach to the embedding calculus knot invariants},

school = {Rheinische Friedrich-Wilhelms-Universität Bonn},

year = 2020,

month = oct,

note = {In this thesis we consider two homotopy theoretic approaches to the study of spaces of knots: the theory of finite type invariants of Vassiliev and the embedding calculus of Goodwillie and Weiss, and address connections between them.

Our results confirm that the knot invariants ev

_{n}produced by the embedding calculus for (long) knots in a 3-manifold M are surjective for all n ≥ 1. On one hand, this solves certain remaining open cases of the connectivity estimates of Goodwillie and Klein, and on the other hand, confirms a part of the conjecture by Budney, Conant, Scannell and Sinha that for the case of classical knots evn are universal additive Vassiliev invariants over Z.There are two crucial ingredients for this result. Firstly, we study the so-called Taylor tower of the embedding calculus more generally for long knots in any manifold with dim(M) ≥ 3 and develop a geometric understanding of its layers (fibres between two consecutive spaces in the tower). In particular, we describe their first non-vanishing homotopy groups in terms of groups of decorated trees.

Secondly, we give an explicit interpretation of ev

_{n}when dim(M)=3 using capped grope cobordisms. These objects were introduced by Conant and Teichner in a geometric approach to the finite type theory, but turn out to exactly describe certain points in the layers.Our main theorem then states that the first possibly non-vanishing embedding calculus invariant of a knot which is grope cobordant to the unknot is precisely the equivalence class of the underlying decorated tree of the grope in the homotopy group of the layer. The surjectvity of ev

_{n}onto the components of the Taylor tower follows from this immediately.As another corollary we obtain a sufficient condition for the mentioned conjecture to hold over a certain coefficient group A. Namely, it is enough that the spectral sequence for the homotopy groups of the Taylor tower, tensored with A, collapses along the diagonal. In particular, such a collapse result is known for A = Q, confirming that the embedding calculus invariants are universal rational additive Vassiliev invariants, and that they factor configuration space integrals through the Taylor tower. It also follows that they are universal over the p-adic integers in a range depending on the prime p, using recent results of Boavida de Brito and Horel.

Moreover, the surjectivity of ev

_{n}implies that any two group structures on the path components of the tower, which are compatible with the connected sum of knots, must agree.Finally, we also discuss the geometric approach to the finite type theory in terms of the Gusarov–Habiro filtration of the set of isotopy classes of knots in a 3-manifold. We extend some known techniques to prove that the associated graded quotients of this filtration are abelian groups, and study the map which relates these groups to certain graph complexes.},

url = {http://hdl.handle.net/20.500.11811/8651}

}