118 C. Cerf and A. Stasiak
aOdd,bEven
The closure of this tangle gives rise to a family of knots withnpositive cross-
ings:ain the vertical row andbin the horizontal row. Looking at Fig. 6.3b,
we see that we may nullify one positive crossing from the vertical row and
b−1 positive crossings from the horizontal row. Thuswx=1+(b−1) =b
andwy=(a−1) + 1 =a. Using (6.1) and the fact thata+b=n,wehave:
PWr=10
7
b+4
7
a=
4
7
(a+b)+6
7
b=
4
7
n+6
7
b. (6.5)Notice that whenb= 2, we get the family of odd twist knots (5 2 , 72 , 92 ,...)
and in that case
PWr=4
7
n+12
7
. (6.6)
aEven,bEven
The closure of this tangle gives rise to a family of knots withapositive cross-
ings in the vertical row andbnegative crossings in the horizontal row. Fig-
ure 6.3c shows that we may nullify only one positive crossing from the vertical
row and one negative crossing from the horizontal row. Thuswx=1−1=0
andwy=(a−1)−(b−1) =a−b. (6.1) gives:
PWr=4
7
(a−b). (6.7)Notice that whenb = 2, we get the family of even twist knots (4 1 , 61 , 81 ,...).
In that case,a=n−2and
PWr=4
7
((n−2)−2)=4
7
n−16
7
. (6.8)
aEven,bodd
The closure of this tangle gives rise to a family of knots withnnegative cross-
ings:ain the vertical row andbin the horizontal row. Looking at Fig. 6.3d,
we see that we may nullifya−1 negative crossings from the vertical row and