Homomorphic expansions for knotted trivalent graphs

TL;DR

This paper introduces a modified universal finite type invariant for knotted trivalent graphs that is homomorphic under a broad set of operations, enhancing algebraic and topological analysis of knots.

Contribution

It develops a new expansion for knotted trivalent graphs that is homomorphic with respect to key operations, improving upon previous invariants.

Findings

The new invariant is homomorphic under a large set of operations.

It maintains connections with Drinfel'd associators and algebraic knot theory.

Provides a simple proof of the LMO invariant's behavior under Kirby II move.

Abstract

It had been known since old times [MO, Da] that there exists a universal finite type invariant ("an expansion") Z^{old} 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 Z^{old} 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 Z^{old} 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 [LMMO].