Embedded graph invariants in Chern-Simons theory

Chern-Simons gauge theory, since its inception as a topological quantum fie ld theory, has proved to be a rich source of understanding for knot invaria nts. In this work the theory is used to explore the definition of the expec tation value of a network of Wilson lines - an embedded graph invariant. Us ing a generalization of the variational method, lowest-order results for in variants for graphs of arbitrary valence and general vertex tangent space s tructure are derived. Gauge invariant operators are introduced. Higher orde r results are found. The method used here provides a Vassiliev-type definit ion of graph invariants which depend on both the embedding of the graph and the group structure of the gauge theory. It is found that one need not fra me individual vertices. However, without a global projection of the graph t here is an ambiguity in the relation of the decomposition of distinct verti ces. It is suggested that framing may be seen as arising from this ambiguit y - as a way of relating frames at distinct vertices. (C) 1999 Elsevier Sci ence B.V. All rights reserved.