Topology in Molecular Biology

6 Writhe Versus the Number of Crossings 123

more negative crossings of the same type. The predicted writhe is unchanged,
leading to a slope of 0 versusn. An example is given by the family composed of
knots with tangles (3)(1)(2),(3)(3)(4),(3)(5)(6),...whose members all have
PWr=20/7. Notice that we do not examine cases of coordinated changes
in rows by steps of one crossing at a time, because these steps may convert
a row of crossings of twist type into a row of crossings of torus type or vice
versa.

6.4.2PWrof Achiral Knots

Since the 3D writhe is a measure of chirality of oriented closed curves in 3D
space, it is a good test to see how achiral knots behave when seen as members
of Conway families. The case of 4 1 is interesting. It belongs to the family of
even twist knots (4 1 , 61 , 81 ,...) for which we have seen above (6.8) thatPWr
is a linear function ofnwith slope 4/7:

PWr=

4

7

n−

16

7

.

Let us replacenby 4 and, as by a miracle,PWrbecomes zero! Now, let us
consider knot 4 1 as a member of another Conway family. 4 1 has rational tangle
(2)(2) and 8 3 , another achiral knot, has rational tangle (4)(4). We should thus
be able to express that they belong to a Conway family with slope 0 versus
n(since bothPWrmust be equal to zero). Indeed, both considered tangles
have two rows, one with positive crossings of twist type and one with negative
crossings of twist type (see Fig. 6.3c) so adding two crossings to each row gives
a net result of zero. We are in the case of a coordinated change in several rows
leading to a slope of 0 versusn. The same Conway family contains knots with
tangles (6)(6),(8)(8), etc. all of which are achiral.

6.4.3 Shifts BetweenPWras Linear Functions ofn............

Let us consider family (a)(b),nodd, withbeven and fixed (thusa=n−bis
odd) so by (6.5):

PWr=

4

7

n+

6

7

b

and compare it to family (a)(b),neven, withbeven and fixed to the same
value (thusa=n−bis even) so by (6.7):

PWr=

4

7

(a−b)

=

4

7

(n−b−b)

=

4

7

n−

8

7

b.