Volume 16, issue 2 (2016)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 23
Issue 9, 3909–4400
Issue 8, 3417–3908
Issue 7, 2925–3415
Issue 6, 2415–2924
Issue 5, 1935–2414
Issue 4, 1463–1934
Issue 3, 963–1462
Issue 2, 509–962
Issue 1, 1–508

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
Other MSP Journals
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

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 Fn: the smallest subgroup of Aut(Fn) 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 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 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 A of chord diagrams for ordinary knotted objects are related to metrized Lie algebras, the spaces Aw 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.

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
Received: 12 April 2015
Revised: 1 July 2015
Accepted: 10 July 2015
Published: 26 April 2016
Dror Bar-Natan
Department of Mathematics
University of Toronto
Toronto ON M5S 2E4
Zsuzsanna Dancso
Mathematical Sciences Institute
Australian National University
John Dedman Building 27
Union Ln
Canberra ACT 2601