singly encircled band with two half-twists is topologically equivalent to a doubly-encircled untwisted band.
- The Linking number describes both forms, and comprises;
Lk = Tw + Wr
Wr or Writhe measures the extent to which coiling of the
central curve has relieved local twisting T of the
ribbon (Tw is ∫T)
- Writhe connects the manifold of a single doubly-twisted band in 2D space (T=2, W=0) with
that of a
doubly-encircled untwisted band in 2D space (T=0, W=2) by projection
into 3D space (T + W = 2)
Both Tw and Wr are signed Chiral indices.
- Positive values of Wr are termed overtwisting by supercoiling
- Negative values of Wr are termed undertwisting by supercoiling
The White, Cãlugãreanu, Fuller Theorem; J. H. White, Am. J. Math., 1969,
91, 693-728; G. Cãlugãreanu,
Czech Mathematics J., 1961, 11, 588-625;
F. Fuller, Proc. Natl. Acad. Sci.,
1971, 68, 815-819.
© H. S. Rzepa, 2007.