Invariants of graphs

Abstract

We address a classical problem in low dimensional topology: the classification of tamely embedded, finite, connected graphs $G$ in $S\sp3$ up to ambient isotopy. In the case that the graph $G$ is homeomorphic to $S\sp1$, our problem reduces to the embedding problem for knots in $S\sp3$. Our major result is the existence of a unique isotopy class of longitudes of a cycle for an infinite class of graphs. We then define new invariants for this infinite class of graphs. First we define a longitude $l\sb{c}$ of a cycle $c$ in $G$. In contrast to the situation of a knot, for a graph it is quite difficult to canonically select an isotopy class of longitudes, since the mapping class group of a many punctured torus is very large. However we prove that longitudes exist for any cycle in any finite graph and are unique in $H\sb1(S\sp3-G;\doubz)$. This definition of a longitude can be considered an extension of the definition of a longitude of a tamely embedded knot in $S\sp3$. We describe the specific conditions under which $l\sb{c}$ is unique in $\Pi$, the fundamental group of the graph complement, as well as the class of graphs which possess a basis of unique longitudes. Next, in the situation in which $l\sb{c}$ is unique for a cycle $c$ in $G$, we define a sequence of invariants $\bar\mu\sb{G}$ which detects whether $l\sb{c}$ lies in $\Pi\sb{n}$, the $n\sp{th}$ term of the lower central series of $\Pi$. These invariants can be viewed as extensions of Milnor's $\bar\mu\sb{L}$ invariants of a link $L$. Although $\bar\mu\sb{G}$ is not a complete invariant, we provide an example illustrating that $\bar\mu\sb{G}$ is more sensitive than Milnor's $\bar\mu\sb{L}$, where L is the subgraph of G consisting of a link.