122 C. Cerf and A. Stasiak
6.4 Discussion .................................................
6.4.1 When isPWra Linear Function ofn? ..................
The simplest case is a family of rational tangles (a)(b)(c)...where alla,b,c,...
are fixed except one. The nonfixed number is equal tonminus a constant
since the sum of alla,b,c,...isn. Therefore in all those casesPWris a linear
function ofnwith slope± 4 /7or± 10 /7.
We can then consider families of rational tangles (a)(b)(c)...where several
ofa,b,c,...change in a coordinated fashion, such thatPWris still a linear
function ofn. Two interesting cases in this regard are slopes±1 and 0. Let
us first point out that when link orientation is indicated on rows of crossings,
two situations occur. Either the arrows are antiparallel (crossings are called
oftwisttype) and only one crossing will be nullified (e.g., rowaon Fig. 6.3b),
or the arrows are parallel (crossings are called oftorustype) and all but one
crossings will be nullified (e.g., rowbon Fig. 6.3b). Each row, in turn, may be
composed of positive or negative crossings. It follows from Formula (6.1) that
a row withxcrossings of twist type will contribute
±
(
10
7
+
4
7
(x−1))
=±
(
1+
4
7
x)
toPWrwhile a row withxcrossings of torus type will contribute
±
(
10
7
(x−1) +4
7
)
=±
(
10
7
x− 1)
toPWr. If we consider a family of rational links where each successive member
has two more positive crossings of twist type in a givenrowand two more
positive crossings of torus type in another row, the increase inPWrwill be
of
4
7
·2+
10
7
·2=4
for an increase innof 4 (four more crossings), leading to a slope of +1. As an
example, the family composed of knots with tangles (3)(2),(5)(4),(7)(6),...
have
PWr=4+4
7
,8+
4
7
,12 +
4
7
,...
respectively. Similarly, if we consider a family of rational links where each
successive member has two more negative crossings of twist type in a given
row and two more negative crossings of torus type in another row,PWrwill
decrease:
−4
7
· 2 −
10
7
·2=− 4 ,
whilenwill increase by 4 (four more crossings), leading to a slope of−1.
Let us now examine a family of rational links where each successive mem-
ber has two more positive crossings of a given type (twist or torus) and two