On the Bishop Invariants of Embeddings of $S^3$ into $\mathbb{C}^3$

The Bishop invariant is a powerful tool in the analysis of real submanifolds of complex space that associates to every (non-degenerate) complex tangent of the embedding a non-negative real number (or infinity). It is a biholomorphism invariant that gives information regarding the local hull of holomorphy of the manifold near the complex tangent. In this paper, we derive a readily applicable formula for the computation of the Bishop invariant for graphical embeddings of 3-manifolds into $\mathbb{C}^3$. We then exhibit some examples over $S^3$ demonstrating the different possible configurations of the Bishop invariant along complex tangents to such embeddings. We will also generate a few more results regarding the behavior of the Bishop invariant in certain situations. We end our paper by analyzing the different possible outcomes from the perturbation of a degenerate complex tangent.