Knots and links without parallel tangents

Abstract

Steinhaus conjectured that every closed oriented $C^1$-curve has a pair of anti-parallel tangents. Porter disproved the conjecture by showing that there exist curves with no anti-parallel tangents. Colin Adams rised the question of whether there exists a nontrivial knot in ${\mathbb{R}}^3$ which has no parallel or antiparallel tangents. The main result of this paper solves this problem, showing that any (smooth or polygonal) link $L$ in ${\mathbb{R}}^3$ is isotopic to a smooth link $\hat{L}$ which has no parallel or antiparallel tangents.

11 pages, 0 figures