#### Volume 16, issue 2 (2016)

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

### Dror Bar-Natan and Zsuzsanna Dancso

Algebraic & Geometric Topology 16 (2016) 1063–1133
##### Abstract

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 “usual” counterparts.

The group of w-braids was studied (under the name “welded braids”) by Fenn, Rimanyi and Rourke and was shown to be isomorphic to the McCool group of “basis-conjugating” automorphisms of a free group ${F}_{n}$: the smallest subgroup of $Aut\left({F}_{n}\right)$ that contains both braids and permutations. Brendle and Hatcher, in work that traces back to Goldsmith, have shown this group to be a group of movies of flying rings in ${ℝ}^{3}$. Satoh studied several classes of w-knotted objects (under the name “weakly-virtual”) and has shown them to be closely related to certain classes of knotted surfaces in ${ℝ}^{4}$. So w-knotted objects are algebraically and topologically  interesting.

Here we study finite-type invariants of w-braids and w-knots. Following Berceanu and Papadima, we construct homomorphic universal finite-type invariants of w-braids. The universal finite-type invariant of w-knots is essentially the Alexander polynomial.

Much as the spaces $\mathsc{A}$ of chord diagrams for ordinary knotted objects are related to metrized Lie algebras, the spaces ${\mathsc{A}}^{w}$ of “arrow diagrams” for w-knotted objects are related to not-necessarily-metrized Lie algebras. Many questions concerning w-knotted objects turn out to be equivalent to questions about Lie algebras. Later in this paper series we re-interpret the work of Alekseev and Torossian on Drinfel’d associators and the Kashiwara–Vergne problem as a study of w-knotted trivalent graphs.

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##### Keywords
virtual knots, w-braids, w-knots, w-tangles, welded knots, knotted graphs, finite-type invariants, Alexander polynomial, Kashiwara–Vergne, associators, free Lie algebras
##### Mathematical Subject Classification 2010
Primary: 57M25, 57Q45
##### Publication
Revised: 1 July 2015
Accepted: 10 July 2015
Published: 26 April 2016
##### Authors
 Dror Bar-Natan Department of Mathematics University of Toronto Toronto ON M5S 2E4 Canada http://www.math.toronto.edu/~drorbn Zsuzsanna Dancso Mathematical Sciences Institute Australian National University John Dedman Building 27 Union Ln Canberra ACT 2601 Australia http://www.math.toronto.edu/zsuzsi