Unlinked Embedded Graphs

This paper is a self-contained development of an invariant of graphs embedded in three-dimensional Euclidean space using the Jones polynomial and skein theory. Some examples of the invariant are computed. An unlinked embedded graph is one that contains only trivial knots or links. Examples show that the invariant is sufficiently powerful to distinguish some different unlinked embeddings of the same graph. The Jones polynomial[J] assigns an invariant to each oriented link L in R which is a Laurent polynomial VL(t) in the variable √ t. The product VL(t)VL(t −1) is another Laurent polynomial in t and can be expressed in terms of λ = t + t−1 to give an ordinary polynomial rL(λ) = rL(t+ t −1) = VL(t)VL(t−1). The polynomial RL(λ) is defined by 1 RL(λ) = (λ+ 2)rL(λ). The invariant RL does not depend on the orientation of the link L and cannot distinguish a link from its mirror image. The advantage of the polynomialRL is that the definition extends to an invariant of graphs embedded in R, a generalisation of the idea of a link in which vertices where a number of edges meet are allowed. The idea of this paper is to develop this invariant using elementary ideas from skein theory and this definition of RL for links. The paper is restricted to the simplest case for which every vertex is four-valent (has degree four), but there is a generalisation to vertices of any even valence. A graph is defined here as a compact polyhedron which is locally isomorphic to either an interval of the real line or to ; in particular a component of a graph may be a circle with no vertices. 1The extra factor λ+ 2 is included so that the unknot U has invariant RU (λ) = λ+ 2 and the empty link has polynomial 1. Typeset by AMS-TEX 1