Dror Bar-Natan
Published 2015-11-17Version 1
In the previous three papers in this series, [WKO1]-[WKO3] (arXiv:1405.1956, arXiv:1405.1955, and to appear), Z. Dancso and I studied a certain theory of "homomorphic expansions" of "w-knotted objects", a certain class of knotted objects in 4-dimensional space. When all layers of interpretation are stripped off, what remains is a study of a certain number of equations written in a family of spaces $A^w$, closely related to degree-completed free Lie algebras and to degree-completed spaces of cyclic words. The purpose of this paper is to introduce mathematical and computational tools that enable explicit computations (up to a certain degree) in these $A^w$ spaces and to use these tools to solve the said equations and verify some properties of their solutions, and as a consequence, to carry out the computation (up to a certain degree) of certain knot-theoretic invariants discussed in [WKO1]-[WKO3] and in my related paper [KBH] (arXiv:1308.1721).
Comments: About 49 pages. The version at http://www.math.toronto.edu/~drorbn/LOP.html#WKO4 may be more recent
Finite Type Invariants of w-Knotted Objects: From Alexander to Kashiwara and Vergne
Finite type invariants of knots via their Seifert matrices
The intersection polynomials of a virtual knot
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