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The Bishop invariant is a powerful tool in the analysis of real submanifoldsof complex space that associates to every (non-degenerate) complex tangent ofthe embedding a non-negative real number (or infinity). It is a biholomorphisminvariant that gives information regarding the local hull of holomorphy of themanifold near the complex tangent. In this paper, we derive a readilyapplicable formula for the computation of the Bishop invariant for graphicalembeddings of 3-manifolds into $\mathbb{C}^3$. We then exhibit some examplesover $S^3$ demonstrating the different possible configurations of the Bishopinvariant along complex tangents to such embeddings. We will also generate afew more results regarding the behavior of the Bishop invariant in certainsituations. We end our paper by analyzing the different possible outcomes fromthe perturbation of a degenerate complex tangent.

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