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

Abstract

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 [FRR] and was shown to be isomorphic to the McCool group [Mc] of "basisconjugating" automorphisms of a free group F n -the smallest subgroup of Aut(F n ) that contains both braids and permutations. Brendle and Hatcher [BH], in work that traces back to Goldsmith [Gol], have shown this group to be a group of movies of flying rings in R 3 . Satoh [Sa] 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.In this article we study finite type invariants of w-brainds and w-knots. Following Berceanu and Papadima [BP], we construct homomorphic universal finite type invariants of w-braids. We find that 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, we find that the spaces 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, and in later papers of this series we re-interpret Alekseev-Torossian's [AT] work on Drinfel'd associators and the Kashiwara-Vergne problem as a study of w-knotted trivalent graphs.

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

“…In the first paper [WKO1], we took a classical approach to studying finite type invariants of w-braids and w-knots and proved that the universal finite type invariant for w-knots is essentially the Alexander polynomial. In this paper we will study finite type invariants of w-tangles and w-tangled foams from a more algebraic point of view, and prove that "homomorphic" universal finite type invariants of w-tangled foams are in one-to-one correspondence with solutions to the (Alekseev-Torossian version of) the Kashiwara-Vergne problem in Lie theory.…”

“…In this paper we will study finite type invariants of w-tangles and w-tangled foams from a more algebraic point of view, and prove that "homomorphic" universal finite type invariants of w-tangled foams are in one-to-one correspondence with solutions to the (Alekseev-Torossian version of) the Kashiwara-Vergne problem in Lie theory. Mathematically, this paper does not depend on the results of [WKO1] in any significant way, and the reader familiar with the theory of finite type invariants will have no difficulty reading this paper without having read [WKO1]. However, since this paper starts with an abstract rephrasing of the well-known finite type story in terms of general algebraic structures, readers who need an introduction to finite type invariants may find it more pleasant to read [WKO1] first (especially Sections 1, 2 and 3.1-3.6).…”

“…Mathematically, this paper does not depend on the results of [WKO1] in any significant way, and the reader familiar with the theory of finite type invariants will have no difficulty reading this paper without having read [WKO1]. However, since this paper starts with an abstract rephrasing of the well-known finite type story in terms of general algebraic structures, readers who need an introduction to finite type invariants may find it more pleasant to read [WKO1] first (especially Sections 1, 2 and 3.1-3.6).…”

“…Motivation and hopes. This article and its siblings [WKO1] and [WKO3] are efforts towards a larger goal. Namely, we believe many of the difficult algebraic equations in mathematics, especially those that are written in graded spaces, more especially those that are related in one way or another to quantum groups [Dr1], and to the work of Etingof and Kazhdan [EK], can be understood, and indeed would appear more natural, in terms of finite type invariants of various topological objects.…”

“…In [WKO1] we studied the universal finite type invariants of w-braids and w-knots, the latter of which turns out to be essentially the Alexander polynomial. A more thorough introduction about our "hopes and dreams" and the u-v-w big picture can also be found in [WKO1].…”