A knot property, also called the twist number, defined as the sum of crossings 
of a link 
,
 |
(1)
|
where 
defined to be 
if the overpass slants from top left to bottom right or
bottom left to top right and 
is the set of crossings of an oriented link.
The writhe of a minimal knot diagram is not a knot invariant, as exemplified by the Perko pair, which have differing writhes (Hoste et al. 1998). This is because while the writhe is invariant under Reidemeister moves II and III, it may increase or decrease by one for a Reidemeister move of type I (Adams 1994, p. 153).
Thistlethwaite (1988) proved that if the writhe of a reduced alternating projection of a knot is not 0, then the knot is not amphichiral (Adams 1994).
A formula for the writhe is given by
 |
(2)
|
where 
is parameterized by 
for 
along the length of the knot by parameter 
, and the frame 
associated with 
is
 |
(3)
|
where 
is a small parameter, 
is a unit vector field
normal to the curve at 
,
and the vector field 
is given by
 |
(4)
|
(Kaul 1999).
Letting Lk be the linking number of the two components of a ribbon, Tw be the twist, and Wr be the writhe, then the Calugareanu theorem states that
 |
(5)
|
(Adams 1994, p. 187).
Knots." Math. Intell. 20, 33-48,
Fall 1998.Kauffman, L.