Abstract
A topological proper knot is a proper embedding f:ℝ1→M3 of the real line into an open 3-manifold. Two proper knots are equivalent if they can be connected by a topological proper isotopy. In this paper, we answer a question posed by the author in [6] and show that, up to topological equivalence and orientation, all proper knots running between the opposite ends of D2×ℝ1 are equivalent. Then sufficient conditions for a topological proper knot to be equivalent to a piecewise linear proper knot are given.
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