Cochran, Tim D.2009-06-042009-06-041996Ghuman, Simrat M.. "Invariants of graphs." (1996) Diss., Rice University. <a href="https://hdl.handle.net/1911/16976">https://hdl.handle.net/1911/16976</a>.https://hdl.handle.net/1911/16976We 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.82 p.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.MathematicsThesisThesis Math. 1996 Ghuman