Some almost polyhedral wild arcs

Alford, W. R.; Ball, B. J.

doi:10.1215/s0012-7094-63-03004-7

Cited by 21 publications

(16 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It would appear that the arcs An of Afford and Ball [1] have these properties, although I can find no proof of this. Remark.…”

Section: Figurementioning

confidence: 88%

On the Nullity and Enclosure Genus of Wild Knots

McPherson¹

1969

Transactions of the American Mathematical Society

View full text Add to dashboard Cite

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.

show abstract

“…It would appear that the arcs An of Afford and Ball [1] have these properties, although I can find no proof of this. Remark.…”

Section: Figurementioning

confidence: 88%

On the Nullity and Enclosure Genus of Wild Knots

McPherson¹

1969

Transactions of the American Mathematical Society

View full text Add to dashboard Cite

show abstract

“…Alford and Ball [1] give examples of two-spheres with one wild point and penetration index of any desired positive odd integer. They form these by joining together arcs as in Figures 3 and 4.…”

Section: (M)->π 2 (N)is Nontrivial So Ndf(m)mentioning

confidence: 99%

A homeomorphism classification of wildly imbedded two-spheres inS³

Sinal¹

1978

Pacific J. Math.

View full text Add to dashboard Cite

show abstract

“…Do there exist uncountably many arcs in Euclidean 3-space R z , which are tame modulo one endpoint? ( [1], p. 33. Such arcs are called "nearly polyhedral" in [1].)…”

mentioning

confidence: 99%

“…Figure 2(a) shows a Fox-Artin arc, which may be constructed from an infinite family of cylinders of the type shown in Figure 2(a) (cf. [3] or [1] for details). For each ί, the link L(E i9 E i+1 ) is the link…”

mentioning

confidence: 99%

Wild arcs in three-space. I. Families of Fox-Artin arcs

McPherson¹

1973

Pacific J. Math.

View full text Add to dashboard Cite

Some almost polyhedral wild arcs

Cited by 21 publications

References 0 publications

On the Nullity and Enclosure Genus of Wild Knots

On the Nullity and Enclosure Genus of Wild Knots

A homeomorphism classification of wildly imbedded two-spheres inS³

Wild arcs in three-space. I. Families of Fox-Artin arcs

Contact Info

Product

Resources

About

Some almost polyhedral wild arcs

Cited by 21 publications

References 0 publications

On the Nullity and Enclosure Genus of Wild Knots

On the Nullity and Enclosure Genus of Wild Knots

A homeomorphism classification of wildly imbedded two-spheres inS3

Wild arcs in three-space. I. Families of Fox-Artin arcs

Contact Info

Product

Resources

About

A homeomorphism classification of wildly imbedded two-spheres inS³