Finite Type Invariants of w-Knotted Objects II: Tangles, Foams and the Kashiwara-Vergne Problem

Bar-Natan, Dror; Dancso, Zsuzsanna

doi:10.48550/arxiv.1405.1955

Cited by 4 publications

(38 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…via the "associated graded" procedure described in [WKO2]. Here K w pSq is the set of S-labelled w-tangles [WKO2], K w pH; T q is the set of w-knotted H-labelled hoops and Tlabelled balloons [BN4], K w pS; Sq is the same but with H " T " S, and δ is the same as in [BN4].…”

Section: Z Zmentioning

confidence: 99%

“…In Lie theory, they represent "universal" g-invariant tensors in U pIgq bS , where Ig :" g ġ˚5 and g is some finite dimensional Lie algebra ([WKO1]- [WKO3]). Readers of Alekseev and Torossian [AT] may care about A w because using notation from [AT], A w pÒ n q is the completed universal enveloping algebra of pa n ' tder n q ˙tr n (see [WKO2]), and hence much of the [AT] story can be told within A w . Several significant Lie theoretic problems (e.g., the Kashiwara-Vergne problem, [KV, AT,WKO2]) can be interpreted as problems about A w pÒ n q.…”

Section: Atmentioning

confidence: 99%

“…We denote by dA the operation of likewise flipping (with signs) all strands: dA " dA S :" ś aPS dA a . In topology, dA a is the reversal of the 1D orientation of a knotted tube [WKO2]. In Lie theory, it is the antipode of UpIgq combined with the sign reversal ϕ Ñ ´ϕ acting on the g ˚factor of Ig.…”

Section: Atmentioning

confidence: 99%

“…(5) Similarly, D dS a is the result of "flipping over stand a and multiplying by a p´q sign for each arrow head or tail that connects to a", as illustrated above 7 . In topology, dS a is the reversal of both the 1D and the 2D orientation of a knotted tube [WKO2]. In Lie theory, it is the antipode of UpIgq.…”

Section: Atmentioning

confidence: 99%

“…(3) The papers [WKO1,WKO2] (and likewise [BN4]) are about equations, but even more so, about the construction of certain knot and tangle invariants. With the tools presented computations below…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Finite Type Invariants of w-Knotted Objects IV: Some Computations

Bar-Natan¹

2015

Preprint

Self Cite

View full text Add to dashboard Cite

In the previous three papers in this series, [WKO1]-[WKO3], Z. Dancso and I studied a certain theory of "homomorphic expansions" of "w-knotted objects", a certain class of knotted objects in 4-dimensional space. When all layers of interpretation are stripped off, what remains is a study of a certain number of equations written in a family of spaces A w , closely related to degree-completed free Lie algebras and to degree-completed spaces of cyclic words.The purpose of this paper is to introduce mathematical and computational tools that enable explicit computations (up to a certain degree) in these A w spaces and to use these tools to solve the said equations and verify some properties of their solutions, and as a consequence, to carry out the computation (up to a certain degree) of certain knot-theoretic invariants discussed in [WKO1]-[WKO3] and in my related paper [BN4]. Contents 1. Introduction 1.1. Acknowledgement 2. Group-like elements in A w 2.1. A brief review of A w 2.2. Some preliminaries about free Lie algebras and cyclic words 2.3. The lower-interlaced presentation E l of A w exp 2.4. The factored presentation E f of A w exp and its stronger precursor E s 2.4.1. The family tA w pH; T qu 2.4.2. Operations on tA w pH; T qu. 2.4.3. Group-like elements in tA w pH; T qu. 2.4.4. The inclusion tA w pSqu ãÑ tA w pH; T qu. 2.5. Converting between the E l and the E f presentations. 3. Some Computations 3.1. Tangle Invariants 3.1.1. The General Framework 3.1.2. The Knot 8 17 and the Borromean Tangle 3.2. Solutions of the Kashiwara-Vergne Equations 3.3. The involution τ and the Twist Equation

show abstract

Section: Z Zmentioning

confidence: 99%

Section: Atmentioning

confidence: 99%

Section: Atmentioning

confidence: 99%

Section: Atmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Finite Type Invariants of w-Knotted Objects IV: Some Computations

Bar-Natan¹

2015

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Virtual tangles and fiber functors

Brochier

2019

J. Knot Theory Ramifications

View full text Add to dashboard Cite

We define a category vT of tangles diagrams drawn on surfaces with boundaries. On the one hand we show that there is a natural functor from the category of virtual tangles to vT which induces an equivalence of categories. On the other hand, we show that vT is universal among ribbon categories equipped with a strong monoidal functor to a symmetric monoidal category. This is a generalization of the Shum-Reshetikhin-Turaev theorem characterizing the category of ordinary tangles as the free ribbon category. This gives a straightforward proof that all quantum invariants of links extends to framed oriented virtual links. This also provides a clear explanation of the relation between virtual tangles and Etingof-Kazhdan formalism suggested by Bar-Natan. We prove a similar statement for virtual braids, and discuss the relation between our category and knotted trivalent graphs.

show abstract

Ribbon 2-Knots, $1+1=2$, and Duflo's Theorem for Arbitrary Lie Algebras

Bar-Natan,

Dancso,

Scherich

2018

Preprint

Self Cite

View full text Add to dashboard Cite

We explain a direct topological proof for the multiplicativity of Duflo isomorphism for arbitrary finite dimensional Lie algebras, and derive the explicit formula for the Duflo map. The proof follows a series of implications, starting with "the calculation 1+1=2 on a 4D abacus", using the study of homomorphic expansions (aka universal finite type invariants) for ribbon 2-knots, and the relationship between the corresponding associated graded space of arrow diagrams and universal enveloping algebras. This complements the results of the first author, Le and Thurston, where similar arguments using a "3D abacus" and the Kontsevich Integral were used to deduce Duflo's theorem for metrized Lie algebras; and results of the first two authors on finite type invariants of w-knotted objects, which also imply a relation of 2-knots with the Duflo theorem in full generality, though via a lengthier path.(∆ for doubling) of 2-knots in R 4 , which then, using an appropriate replacement of the Kontsevich integral, becomes an equality of arrow diagrams, which in itself can be interpreted as an equality in (a completion of) S(g * ) g ⊗ S(g * ) g ⊗ U (g) for an arbitrary finite-dimensional Lie algebra g, proving the multiplicative property of the Duflo isomorphism in full generality.

show abstract

Finite Type Invariants of w-Knotted Objects II: Tangles, Foams and the Kashiwara-Vergne Problem

Cited by 4 publications

References 0 publications

Finite Type Invariants of w-Knotted Objects IV: Some Computations

Finite Type Invariants of w-Knotted Objects IV: Some Computations

Virtual tangles and fiber functors

Ribbon 2-Knots, $1+1=2$, and Duflo's Theorem for Arbitrary Lie Algebras

Contact Info

Product

Resources

About