6 Writhe Versus the Number of Crossings 1191 negative crossing from the horizontal row. Thuswx=−(a−1)−1=−a
andwy=− 1 −(b−1) =−b. Using (6.1) and the fact thata+b=n,wehave:PWr=−10
7
a−4
7
b=−4
7
(a+b)−6
7
a=−4
7
n−6
7
a. (6.9)aOdd,bOddFigure 6.3e shows that the closure of such a tangle gives rise to a two-
component link. There are therefore two possible orientations for the second
component.
(a) With the first possible orientation we get links withnpositive crossings:
ain the vertical row andbin the horizontal row. Looking at Fig. 6.3f, we
see that we may nullify 1 positive crossing from the vertical row andb− 1
positive crossings from the horizontal row, like in the case whereais odd
andbeven (case B.1). Thuswx=1+(b−1) =bandwy=(a−1) + 1 =a.
Using (6.1) and the fact thata+b=n, we have, as in case B.1,PWr=10
7
b+4
7
a=4
7
(a+b)+6
7
b=4
7
n+6
7
b. (6.10)Notice that (6.5) and (6.10) are the same, but the first one (case B.1)
deals with knots while we are now dealing with two-component links.
(b) With the other orientation for the second component, we get links with
nnegative crossings:ain the vertical row andbin the horizontal row.
Figure 6.3g shows that we may nullifya−1 negative crossings from the
vertical row and 1 negative crossing from the horizontal row, like in the
case whereais even andbodd (case B.3). Thuswx=−(a−1)−1=−a
andwy=− 1 −(b−1) =−b. Using (6.1) and the fact thata+b=n,we
have, as in case B.3,
PWr=−10
7
a−4
7
b=−4
7
(a+b)−6
7
a=−4
7
n−6
7
a. (6.11)