Topology in Molecular Biology

5 From Tangle Fractions to DNA 93

Note that we only need to take numerator closures, since the denominator
closure of a tangle is simply the numerator closure of its rotate.
As already mentioned in the introduction, it may happen that two rational
tangles are non-isotopic but have isotopic numerators. The simplest instance of
this phenomenon is addingntwists at the bottom of a tangleT, see Fig. 5.18.
This operation does not change the knotN(T),i.e.N(T∗ 1 /[n])∼N(T),but
it does change the tangle, sinceF(T∗ 1 /[n]) =F(1/([n]+1/T)) = 1/(n+
1 /F(T)); so, ifF(T)=p/q, thenF(T∗ 1 /[n]) =p/(np+q).Hence, if we set
np+q=q′we haveq≡q′(modp),just as Theorem 2 dictates. Note that
reducing all possible bottom twists implies|p|>|q|.
Another key example of the arithmetic relationship of the classification of
rational knots is illustrated in Fig. 5.19. Here we see that the “palindromic”
tangles

T= [[2],[3],[4]] = [2] +

1

[3] +[4]^1

T T

N(T) ~ N(T ) *

[n]

_^1

[n]

_^1

~

Fig. 5.18.Twisting the bottom of a tangle

T = [2] + 1/( [3] + 1/[4] ) S = [4] + 1/( [3] + 1/[2] )

~

N(T) = N(S)

Fig. 5.19.An instance of the palindrome equivalence