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

James Murdoch McPherson

Vol. 45 (1973), No. 2, 585–598

DOI: 10.2140/pjm.1973.45.585

Abstract

Roughly speaking, a Fox-Artin arc is an arc which is tame modulo one endpoint at which it has penetration index three, and which may be constructed in the way that the examples of R. H. Fox and E. Artin were constructed in their classical paper of 1948.

For each oriented Fox-Artin arc, there is an associated infinite sequence of oriented prime 2-component links, which is an invariant of the local embedding type of the arc in R⁸. Using existence results from link theory, this result yields the corollary: If M is a 3-manifold and p a point in the interior of M, then there exists an uncountable family of locally non-invertible Fox-Artin arcs in M, which are wild at p.

Later papers will be concerned with developing invariants of the oriented local embedding type of an arc k_n which is tame modulo one endpoint, at which it has penetration index 2n + 1.

Mathematical Subject Classification

Primary: 57A10

Secondary: 55A30

Milestones

Received: 23 October 1970

Revised: 5 April 1971

Published: 1 April 1973