Subspaces of knot spaces (Q2723512)

The paper is concerned with the writhe of smoothly embedded knots in \(\mathbb{R}^3\), which is a real number defined by a differential geometric integral as used in work on DNA, rather than the combinatorially defined integer writhe of a knot diagram. The result proved here is that the inclusion of the space of all knots of a fixed writhe in an isotopy class into the space of all knots in that class is a weak homotopy equivalence. The proof depends on modification of families of embeddings parametrised by \(S^n\), making careful use of helical segments to allow suitable control of the writhe.