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Homomorphic Expansions for Knotted Trivalent Graphs

Dror Bar-Natan and Zsuzsanna Dancso

Journal of Knot Theory and Its Ramifications 22-1 (2013)
last updated Fri, 28 Sep 2012 17:34:41 -0400
first edition: March 9, 2011

Abstract. It had been known since old times that there exists a universal finite type invariant ("an expansion") Zold for Knotted Trivalent Graphs (KTGs), and that it can be chosen to intertwine between some of the standard operations on KTGs and their chord-diagrammatic counterparts (so that relative to those operations, it is "homomorphic"). Yet perhaps the most important operation on KTGs is the "edge unzip" operation, and while the behavior of Zold under edge unzip is well understood, it is not plainly homomorphic as some "correction factors" appear.
In this paper we present two (equivalent) ways of modifying Zold into a new expansion Z, defined on "dotted Knotted Trivalent Graphs" (dKTGs), which is homomorphic with respect to a large set of operations. The first is to replace "edge unzips" by "tree connect sums", and the second involves somewhat restricting the circumstances under which edge unzips are allowed. As we shall explain, the newly defined class dKTG of knotted trivalent graphs retains all the good qualities that KTGs have - it remains firmly connected with the Drinfel'd theory of associators and it is sufficiently rich to serve as a foundation for an "Algebraic Knot Theory". As a further application, we present a simple proof of the good behavior of the LMO invariant under the Kirby II (band-slide) move.

The paper. ktgs.pdf (source: ktgs.zip), arXiv:1103.1896.