• Corpus ID: 15224868

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

@article{BarNatan2015FiniteTI,
  title={Finite Type Invariants of w-Knotted Objects IV: Some Computations},
  author={Dror Bar-Natan},
  journal={arXiv: Geometric Topology},
  year={2015}
}
  • D. Bar-Natan
  • Published 17 November 2015
  • Mathematics
  • arXiv: Geometric Topology
In the previous three papers in this series, [WKO1]-[WKO3] (arXiv:1405.1956, arXiv:1405.1955, and to appear), 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… 
View PDF on arXiv
8 Citations
Highly Influential Citations
1
Background Citations
4

Figures from this paper

8 Citations

Finite-type invariants of w-knotted objects, I: w-knots and the Alexander polynomial

This is the first in a series of papers studying w-knots, and more generally, w-knotted objects (w-braids, w-tangles, etc). These are classes of knotted objects which are wider, but weaker than their…

Finite Type Invariants of w-Knotted Objects: From Alexander to Kashiwara and Vergne

This preprint was split in two and became the first two parts of a four-part series (arXiv:1405.1956, arXiv:1405:1955, and two in preparation). The remaining relevance of this preprint is due to the…

Balloons and Hoops and their Universal Finite-Type Invariant, BF Theory, and an Ultimate Alexander Invariant

Balloons are 2D spheres. Hoops are 1D loops. Knotted balloons and hoops (KBH) in 4-space behave much like the first and second homotopy groups of a topological space—hoops can be composed as in π1,…

On usual, virtual and welded knotted objects up to homotopy

We consider several classes of knotted objects, namely usual, virtual and welded pure braids and string links, and two equivalence relations on those objects, induced by either self-crossing changes…

Towards a Version of Markov’s Theorem for Ribbon Torus-Links in $$\mathbb {R}^4$$

  • C. Damiani
  • Mathematics
    Knots, Low-Dimensional Topology and Applications
  • 2019
In classical knot theory, Markov's theorem gives a way of describing all braids with isotopic closures as links in $\mathbb{R}^3$. We present a version of Markov's theorem for extended loop braids…

Milnor Invariants, $2n$-moves and $V^{n}$-moves for Welded String Links

In a previous paper, the authors proved that Milnor link-homotopy invariants modulo $n$ classify classical string links up to $2n$-move and link-homotopy. As analogues to the welded case, in terms of…

Over then Under Tangles

Brilliant wrong ideas should not be buried and forgotten. Instead, they should be mined for the gold that lies underneath the layer of wrong. In this paper we explain how "over then under tangles"…

A Markov's theorem for extended welded braids and links

Extended welded links are a generalization of Fenn, Rim\'{a}nyi, and Rourke's welded links. Their braided counterpart are extended welded braids, which are closely related to ribbon braids and loop…

References

SHOWING 1-10 OF 38 REFERENCES

FINITE TYPE INVARIANTS OF W-KNOTTED OBJECTS II: THE DOUBLE TREE CONSTRUCTION

In this paper we provide a topological interpretation and independent topological proof of the formula by Alkeseev-Enriquez-Torossian AlekseevEnriquezTorossian:ExplicitSolutions [AET] for solutions…

Finite-type invariants of w-knotted objects, I: w-knots and the Alexander polynomial

This is the first in a series of papers studying w-knots, and more generally, w-knotted objects (w-braids, w-tangles, etc). These are classes of knotted objects which are wider, but weaker than their…

Finite type invariants of w-knotted objects II: tangles, foams and the Kashiwara–Vergne problem

This is the second in a series of papers dedicated to studying w-knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.). These are classes of knotted objects that are wider but…

The algebra of knotted trivalent graphs and Turaev's shadow world

Knotted trivalent graphs (KTGs) form a rich algebra with a few simple operations: connected sum, unzip, and bubbling. With these operations, KTGs are generated by the unknotted tetrahedron and Mobius…

Finite Type Invariants of w-Knotted Objects: From Alexander to Kashiwara and Vergne

This preprint was split in two and became the first two parts of a four-part series (arXiv:1405.1956, arXiv:1405:1955, and two in preparation). The remaining relevance of this preprint is due to the…

The Kashiwara-Vergne conjecture and Drinfeld’s associators

The Kashiwara-Vergne (KV) conjecture is a property of the Campbell-Hausdorff series put forward in 1978. It has been settled in the positive by E. Meinrenken and the first author in 2006. In this…

NON-ASSOCIATIVE TANGLES

Following Drinfel’d, Kontsevich and Piunikhin, we study the iterated integral expression for the holonomy of the formal Knizhnik-Zamolodchikov connection, finding that by introducing non-associative…

HOMOMORPHIC EXPANSIONS FOR KNOTTED TRIVALENT GRAPHS

It had been known since old times (works of Murakami–Ohtsuki, Cheptea–Le and the second author) that there exists a universal finite type invariant ("an expansion") Zold for knotted trivalent graphs…

On the Vassiliev knot invariants

Quantization of Lie bialgebras, I

In the paper [Dr3] V. Drinfeld formulated a number of problems in quantum group theory. In particular, he raised the question about the existence of a quantization for Lie bialgebras, which arose…