120 C. Cerf and A. Stasiak
The same remark holds, i.e., (6.9) and (6.11) are identical but the first one
(case B.3) refers to knots while the second one refers to two-component
links.6.3.3 Tangles with Three Rows, Denoted by (a)(b)(c),a,b,andc
Positive Integers
Now,n=a+b+c. Figure 6.4a shows an example of such a tangle, witha=3,
b=1andc=2.Ifweletaincrease by steps of 2 and we fixbandcto 1
and 2, respectively, the closure of these tangles produces the family of knots
62 , 82 , 102 , etc. There are 2^3 = 8 cases to study, depending on the parity ofa,
bandc. Let us illustrate the process withaodd,bodd,ceven. There area
positive crossings in the first horizontal row,bpositive crossings in the vertical
row, andcnegative crossings in the last horizontal row. Looking at Fig. 6.4b,
we see that we may nullifya−1 positive crossings from the first row, 1 positive
crossing from the second row, and 1 negative crossing from the last row. Thus
wx=(a−1) + 1−1=a−1andwy=1+(b−1)−(c−1) = 1 +b−c. Formula
(6.1) gives:
PWr=10
7
(a−1) +4
7
(1 +b−c). (6.12)Sincen=a+b+c,ifb=1andc= 2, thena=n−3 and we get for the
family of knots 6 2 , 82 , 102 , etc.:
PWr=10
7
(n− 3 −1) +4
7
(1 + 1−2)
=
10
7
n−40
7
. (6.13)
6.3.4 Tangles withrRows..................................
We can generalize this approach to any family of rational knots or links.
Formula (6.1) will still hold, where each ofwxandwyis a sum of the following
form:
wx/y=⎧
⎪⎪
⎨
⎪⎪
⎩
a− 1
−(a−1)
1
− 1+
⎧
⎪⎪
⎨
⎪⎪
⎩
b− 1
−(b−1)
1
− 1+
⎧
⎪⎪
⎨
⎪⎪
⎩
c− 1
−(c−1)
1
− 1+···
︸ ︷︷ ︸
rterms