Abstract. Bing-Whitehead Cantor sets were introduced by DeGryse and Osborne in dimension three and greater to produce examples of Cantor sets that were nonstandard (wild), but still had a simply connected complement. In contrast to an earlier example of Kirkor, the construction techniques could be generalized to dimensions greater than three. These Cantor sets in S 3 are constructed by using Bing or Whitehead links as stages in defining sequences. Ancel and Starbird, and separately Wright, characterized the number of Bing links needed in such constructions so as to produce Cantor sets. However it was unknown whether varying the number of Bing and Whitehead links in the construction would produce nonequivalent Cantor sets. Using a generalization of the geometric index, and a careful analysis of three dimensional intersection patterns, we prove that Bing-Whitehead Cantor sets are equivalently embedded in S 3 if and only if their defining sequences differ by some finite number of Whitehead constructions. As a consequence, there are uncountably many nonequivalent such Cantor sets in S 3 constructed with genus one tori and with a simply connected complement.

Abstract. We introduce new general techniques for computing the geometric index of a link L in the interior of a solid torus T . These techniques simplify and unify previous ad hoc methods used to compute the geometric index in specific examples and allow a simple computation of geometric index for new examples where the index was not previously known. The geometric index measures the minimum number of times any meridional disc of T must intersect L. It is related to the algebraic index in the sense that adding up signed intersections of an interior simple closed curve C in T with a meridional disc gives ± the algebraic index of C in T . One key idea is introducing the notion of geometric index for solid chambers of the form B 2 × I in T . We prove that if a solid torus can be divided into solid chambers by meridional discs in a specific (and often easy to obtain) way, then the geometric index can be easily computed.

We construct uncountably many simply connected open 3-manifolds with genus one ends homeomorphic to the Cantor set. Each constructed manifold has the property that any self homeomorphism of the manifold (which necessarily extends to a homeomorphism of the ends) fixes the ends pointwise. These manifolds are complements of rigid generalized Bing-Whitehead (BW) Cantor sets. Previous examples of rigid Cantor sets with simply connected complement in $R^{3}$ had infinite genus and it was an open question as to whether finite genus examples existed. The examples here exhibit the minimum possible genus, genus one. These rigid generalized BW Cantor sets are constructed using variable numbers of Bing and Whitehead links. Our previous result with \v{Z}eljko determining when BW Cantor sets are equivalently embedded in $R^{3}$ extends to the generalized construction. This characterization is used to prove rigidity and to distinguish the uncountably many examples.Comment: arXiv admin note: text overlap with arXiv:0810.343

Abstract. We prove that there exist uncountably many inequivalent rigid wild Cantor sets in R 3 with simply connected complement. Previous constructions of wild Cantor sets in R 3 with simply connected complement, in particular the Bing-Whitehead Cantor sets, had strong homogeneity properties. This suggested it might not be possible to construct such sets that were rigid. The examples in this paper are constructed using a generalization of a construction of Skora together with a careful analysis of the local genus of points in the Cantor sets.

Abstract. We classify the free homotopy classes of closed curves with minimal self intersection number two on a once punctured toms, T, up to homeomorphism. Of these, there are six primitive classes and two imprimitive. The classification leads to the topological result that, up to homeomorphism, there is a unique curve in each class realizing the minimum self intersection number. The classification yields a complete classification of geodesics on hyperbolic T which have self intersection number two. We also derive new results on the Markoff spectrum of diophantine approximation; in particular, exactly three of the impfimitive classes correspond to families of Markoff values below Hall's ray.