Normal holonomy and writhing number of polygonal knots.
The normal holonomy \(Hol\) of a polygonal knot \(K\) is a geometric invariant which is closely related to the writhing number \(Wr\). In fact, in this article it is shown that \(2\pi Wr(K) \equiv Hol(K)\) mod \((2\pi)\) for any polygonal knot \(K\). (The same formula holds for smooth knots.) This says the normal holonomy can be identified with the ``fractional part'' of the writhing number. It is also shown that normal holonomy fibers the space of knots over the circle and that the writhing number fibers the space of knots over the real line. Consequently, two isotopic polygonal knots which have the same writhing number are isotopic through a family of polygonal knots having the same writhing number. Similarly, two isotopic polygonal knots which have zero holonomy are isotopic through a family of polygonal knots having zero holonomy if and only if they have the same autoparallel linking number. In the last section the definition of normal holonomy is extended to immersed polygonal knots (i.e. polygonal knots with possible self intersections). Again it is shown that normal holonomy fibers the space of knots over the circle, but now there are two isotopy classes of immersed knots with zero holonomy.
scientific article; zbMATH DE number 1818985