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The fundamental group of the complement of a wild knot in a 3-sphere can be expressed as the colimit (direct limit) of a suitable family of groups and homomorphisms (Crowell [4]). To each group in the family we assign a Jacobian module, and in §1 we prove that this assignment is functorial and preserves colimits. This is used in §2 to show that the nullity of the Alexander module of a knot with one wild point is bounded above by its enclosure genus. This can be used in some cases to calculate the enclosure genus and the penetration index of the knot in a purely algebraic way. In §3 we give examples to show that the upper bound of Theorem 2 is the best possible, along with some results concerning the penetration index of an arc relative to surfaces of genus r.