Duality properties of indicatrices of knots

TL;DR

This paper explores the duality relationships between various indicatrices of knots, linking geometric invariants like bridge indices to tangent, binormal, and Darboux indicatrices, and extends these concepts to stick knots.

Contribution

It introduces the bridge and inflection maps of knot conformations, interprets them via indicatrices, and studies their duality relationships, including for stick knots.

Findings

Graph of the bridge map is union of binormal indicatrix, antipodal curve, and great circles.

Inflection map relates to the binormal and tangent indicatrices.

Duality between normal and Darboux indicatrices is established.

Abstract

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.