Abstract

The normal holonomy of a polygonal knot is a geometrical invariant which is closely related to the writhing number. We show that normal holonomy fibers the space of knots over the circle and deduce that the writhing number fibers the space of knots over the real line. Consequently, two isotopic knots which have the same writhing number are isotopic through a family of knots having the same writhing number. In a similar vein, two isotopic knots having zero holonomy are isotopic through a family of such knots if and only if they have the same autoparallel linking number.

More generally, the definition of normal holonomy makes sense for immersed polygonal knots. This time normal holonomy fibers the space of immersed knots over the circle, but now there are only two isotopy classes of immersed knots of zero holonomy.

Authors