Finite Type Invariants of w-Knotted Objects I: w-Knots and the Alexander Polynomial

TL;DR

This paper introduces the study of finite type invariants for w-knotted objects, revealing connections to the Alexander polynomial, Lie algebras, and the Kashiwara-Vergne conjecture, thus bridging topology and algebra.

Contribution

It constructs universal finite type invariants for w-knots and relates them to classical invariants and Lie algebra structures, providing new insights into their algebraic and topological properties.

Findings

Universal finite type invariant of w-knots is essentially the Alexander polynomial.

Spaces of arrow diagrams relate to not-necessarily-metrized Lie algebras.

Homomorphic expansion of w-knotted foams links to the Kashiwara-Vergne conjecture.

Abstract

This is the first in a series of papers studying w-knotted objects (w-knots, w-braids, w-tangles, etc.), which make a class of knotted objects which is {w}ider but {w}eaker than their usual counterparts. The group of w-braids was studied (as "{w}elded braids") by Fenn-Rimanyi-Rourke and was shown to be isomorphic to the McCool group of "basis-conjugating" automorphisms of a free group Fn. Brendle-Hatcher, tracing back to Goldsmith, have shown this group to be a group of movies of flying rings in R3. Satoh studied several classes of w-knotted objects (as "{w}eakly-virtual") and has shown them to be closely related to certain classes of knotted surfaces in R4. So w-knotted objects are algebraically and topologically interesting. Here we study finite type invariants of w-knotted objects. Following Berceanu-Papadima, we construct homomorphic universal finite type invariants ("expansions") of w-braids and of w-tangles. We find that the universal finite type invariant of w-knots is essentially the Alexander polynomial. We find that 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. Most notably we find that a homomorphic expansion of w-knotted foams is essentially the same as a solution of the Kashiwara-Vergne conjecture (KV), thus giving a topological explanation to the work of Alekseev-Torossian work on KV and Drinfel'd associators. The true value of w-knots, though, is likely to emerge later, for we expect them to serve as a {w}armup example for the study of virtual knots. We expect v-knotted objects to provide the global context whose associated graded structure will be the Etingof-Kazhdan theory of quantization of Lie bialgebras.