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

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.

arXiv comment52 pages. This paper is part 1 of a 4-part series whose first two parts originally appeared as a combined preprint, arXiv:1309.7155. July 2015: minor mods following a referee report

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• Homotopy classification of ribbon tubes and welded string links
• Arrow calculus for welded and classical links
• Finite Type Invariants of w-Knotted Objects: From Alexander to Kashiwara and Vergne
• The pure braid groups and their relatives
• Pure virtual braids, resonance, and formality
• On Usual, Virtual and Welded knotted objects up to homotopy
• Chen ranks and resonance varieties of the upper McCool groups
• Tangle Machines
• Virtual concordance and the generalized Alexander polynomial
• Finite Type Invariants of w-Knotted Objects IV: Some Computations
• On the welded Tube map
• Milnor invariants of braids and welded braids up to homotopy
• Ribbon 2-Knots, $1+1=2$, and Duflo's Theorem for Arbitrary Lie Algebras
• Taylor expansions of groups and filtered-formality
• A Note on the Unitarity Property of the Gassner Invariant
• A simple solution to the word problem for virtual braid groups
• Milnor invariants, $2n$-moves and $V^n$-moves for welded string links
• A Reidemeister Theorem for Solid Ribbon Torus Links
• Over then Under Tangles