Fox and Harrold first used the
words “Wilder arc” to describe a wild arc in euclidean 3-space E8 which is the union
of two tame arcs meeting only in a common endpoint and which is locally
peripherally unknotted (L.P.U.) at this point of intersection. Thus there are
imposed (a) conditions upon the embeddings of subarcs of the wild arc and (b)
conditions upon the manner in which the subarcs meet. The following definition
gives only conditions of type (a): An arc A ⊂ E3 is almost tame if each
point of A lies on a tame subarc of A. Clearly, every Wilder arc is almost
tame.
The chief result characterizes the set W of points on an almost tame arc at which
the arc can fail to be locally tame. In particular, W is shown to be homeomorphic to
a closed countable set W′ on the unit interval with the property that a point
x ∈ W′ either is the first or last point of W′ or x has either an immediate
predecessor or an immediate successor. Two further results discuss special
cases.
