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.
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Arnold V.I.: The geometry of spherical curves and the algebra of quaternions. Russ. Math. Surv. 50, 1–68 (1995)
Banchoff T.F.: Frenet frames and theorems of Jacobi and Milnor for space polygons. Rad Jugosl. Akad. Znan. Umjet. 396, 101–108 (1982)
Blaschke W.: Vorlesungen über Integralgeometrie, vol. II, 3rd edn. Deutscher Verlag der Wissenschaften, Berlin (1955)
Fáry I.: Sur la courbure totale d’une courbe gauche faisant un nœud. Bull. Soc. Math. France 77, 128–138 (1949)
Fenchel W.: On the differential geometry of closed space curves. Bull. Am. Math. Soc. (N. S.) 57, 44–54 (1951)
Jeon C.B., Jin G.T.: A computation of superbridge index of knots. J. Knot Theory Ramif. 11, 461–473 (2002)
Jin G.T.: Polygon indices and superbridge indices of torus knots and links. J. Knot Theory Ramif. 6, 281–289 (1997)
Kuiper N.H.: A new knot invariant. Math. Ann. 278, 193–209 (1987)
McRae A.S.: The Milnor-Totaro theorem for space polygons. Geom. Dedicata 84, 321–330 (2001)
Milnor J.W.: On the total curvature of knots. Ann. Math. (2) 52, 248–257 (1950)
Milnor J.W.: On total curvatures of closed space curves. Math. Scand. 1, 289–296 (1953)
Schubert H.: Über eine numerische Knoteninvariante. Math. Z. 61, 245–288 (1954)
Uribe-Vargas R.: On singularities, “perestroikas” and differential geometry of space curves. Enseign. Math. (2) 50, 69–101 (2004)
Wu Y.-Q.: Knots and links without parallel tangents. Bull. Lond. Math. Soc. 34, 681–690 (2002)
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Adams, C., Collins, D., Hawkins, K. et al. Duality properties of indicatrices of knots. Geom Dedicata 159, 185–206 (2012). https://doi.org/10.1007/s10711-011-9652-6
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DOI: https://doi.org/10.1007/s10711-011-9652-6