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Duality properties of indicatrices of knots
The bridge index and superbridge index of a knot are important invariants in
knot theory. We define the bridge map of a knot conformation, which is closely
related to these two invariants, and interpret it in terms of the tangent
indicatrix of the knot conformation. Using the concepts of dual and derivative
curves of spherical curves as introduced by Arnold, we show that the graph of
the bridge map is the union of the binormal indicatrix, its antipodal curve,
and some number of great circles. Similarly, we define the inflection map of a
knot conformation, interpret it in terms of the binormal indicatrix, and
express its graph in terms of the tangent indicatrix. This duality relationship
is also studied for another dual pair of curves, the normal and Darboux
indicatrices of a knot conformation. The analogous concepts are defined and
results are derived for stick knots.Comment: 22 pages, 9 figure
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