Intrinsic chirality of graphs in 3-manifolds (Q1738807)

A graph \(\Gamma\) embedded in a 3-manifold \(M\) is chiral if there does not exist an orientation reversing homeomorphism of \(M\) that leaves \(\Gamma\) setwise invariant. A graph is intrinsically chiral if all of its embeddings in \(M\) are chiral. This paper generalizes a sufficient condition for graphs embedded in \(S^3\) to be intrinsically chiral (see [\textit{E. Flapan}, J. Knot Theory Ramifications 4, No. 3, 373--388 (1995; Zbl 0837.57002)]) to all closed, connected, orientable, irreducible 3-manifolds and there are infinitely many graphs that satisfy the generalized condition. Besides, it is proved that there does not exist a graph that is intrinsically chiral in all closed, connected, orientable, irreducible 3-manifolds. These results provide a more comprehensive picture of intrinsic chirality of graphs embedded in 3-manifolds.