On the Nullity and Enclosure Genus of Wild Knots

Abstract

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.

The nullity of a wild knot in a compact 3-manifold

“…Q.E.D. From proposition 2, and theorem 1 of [6], it follows that the Betti module M[G, /JT] of G is the direct limit of the Jacobian modules of the G ; evaluated at a f ; since G is generated by the image groups ip,{Gi), M[G, JST] is the union of its submodules M\]/,{M[G t , aj), and each of these submodules is finitely generated because each G f is. We therefore have a sequence This brings us to the statement of the main theorem.…”

“…We may therefore make G, into a B-group [Gj,a f ] by setting a f = )SC,TJ; then 0 f and i/'j become morphisms in the category 3$ of B-groups, because the diagrams B commute (cf. [6], p. 545). Consider the diagram in which every triangle commutes, with the possible exception of GLB.…”

“…It will be assumed throughout that readers are familiar with the terminology, notation, and results of [1] and [6]; in particular, if G is a group and x: G -> G jG' the canonical map, /? : G jG' ->B the canonical map whose kernel is the torsion subgroup of GIG', then the Jacobian JB-moduIe of G evaluated at fix (cf.…”

“…: G jG' ->B the canonical map whose kernel is the torsion subgroup of GIG', then the Jacobian JB-moduIe of G evaluated at fix (cf. [1], p. 146, and [6], p. 546 f.) will be called the Betti module of G. PROPOSITION…”

Crossed complexes and chain complexes with operators

“…The existence of an adjoint was suggested by results in [25] that the Alexander module preserves colimits. Special cases of the groupoid E 1 = (ΘL) 1 appear in [10], [17] and [16].…”