A polynomial invariant of graphs in 3-manifolds (Q1208257)

Let \(A\) be a finite set of disjoint arcs in a 3-manifold \(M\). For \(a_ i\in A\) let \(M_ i\) be the complement of a regular neighborhood of \(a_ i\). A polynomial in \(x\) and \(y\) is defined recursively: if \(A=\emptyset\), \(P(M,A)=0\) if \(M\) contains an essential 2-sphere, \(\partial M\) is compressible, \(\partial M\) is a 2-sphere, or \(\partial M=\emptyset\); otherwise \(P(M,A)=1\). If \(A\neq\emptyset\), \(P(M,A)=xP(M_ i,A-a_ i)+yP(M,A-a_ i)\) for any \(a_ i\in A\). If \(G\) is a finite graph in a 3-manifold, \(Q(G)\) is defined by \(Q(G)=P(M\) -- neighborhood of the vertices, edges). Specializing to \(S^ 3\), the paper explores the relationship between the polynomial \(Q(G)\) and reducibility, decomposability, and planarity of the graphs \(G\) embedded in \(S^ 3\). Different polynomials can be associated with graphs by changing the definition of the base case \(A=\emptyset\) and using the same recursive definition for other cases. A polynomial \(R(M,a)\) with variables \(x\), \(y\), and \(q\) is defined with the base case: if \(A=\emptyset\), let \(M'\) be obtained from \(M\) by capping off all 2-sphere boundary components; if \(M'\) is prime define \(R(M,\emptyset)=1\) if \(\partial M'\) is compressible or if \(\partial M'=\emptyset\), and \(q\) otherwise. If \(M'\) is not prime let \(R(M,\emptyset)\) be the product over all prime factors \(M''\) of \(M'\) of \(R(M'',\emptyset)\). \(R\) is used to define \(S(G)\) is the same way that \(P\) defined \(Q\). It is then shown that if \(G\) is an abstractly planar graph (homeomorphic to a graph in \(S^ 2\)) in \(S^ 3\) then \(G\) is planar (lies on an imbedded surface homeomorphic to \(S^ 2\)) if and only if \(S(G)\) has no \(q\)'s.