124 C. Cerf and A. Stasiak
The two linear functions ofnare thus shifted by 2b. Sincebis an even
integer, the shift is a multiple of 4. To illustrate this, the linear function
corresponding to the family of odd twist knots (5 2 , 72 , 92 ,...) with tangle
(n−2)(2),nodd, and the linear function corresponding to the family of
even twist knots (4 1 , 61 , 81 ,...) with tangle (n−2)(2),neven, are shifted by
2 ×2 = 4. It is worth noticing that the abscissas of the points on both lines
are different: oddn’s for the first line, evenn’s for the second line.
We may similarly compute the shift between any two linear functions ofn
having the same slope, and this can happen even if the families do not have
the same number of rows in their corresponding tangles. For example, let us
consider the family of torus knots with tangle (a),a=nodd (not fixed), so
by (6.2):
PWr=10
7
n−6
7
,
and compare it to family (a)(b)(c),neven, withbodd and fixed,ceven and
fixed (thusa=n−b−cis odd) so by (6.12):
PWr=10
7
(a−1) +4
7
(1 +b−c)=10
7
(n−b−c−1) +4
7
(1 +b−c)=10
7
n−6
7
b− 2 c−6
7
.
The shift is of 6/ 7 b+2c. To take a specific case, if we compare the family of
torus knots to the family of knots 6 2 , 82 , 102 ,...having tangle (n−3)(1)(2),
(6.13) gives:
PWr=10
7
n−40
7
,
and the shift is of 10/7+4=34/7.
6.4.4 Knots Versus Two-Component Links
As already mentioned, the closure of a rational tangle gives rise either to a
knot or to a two-component link. Actually, this can be directly related to the
parity of the nullification writhewx(see Prop. 12 of [9]). For a family of knots
wxis even while for a family of two-component linkswxis odd.
6.5 Conclusion
Using the formulaPWr=10/ 7 wx+4/ 7 wyintroduced in [8], we can pre-
dict the 3D writhe of any rational knot or link in its ideal configuration,
or equivalently, the ensemble average of the 3D writhe of random configu-
rations of it. For every family of knots or links corresponding to a rational