Topology in Molecular Biology

5 From Tangle Fractions to DNA 89

Atδ= 0 we also have:

〈N([0]〉 =0,〈D([0])〉 =1,〈N([∞])〉 =1,〈D([∞])〉=0,and so, the
evaluations 3–5 are easy. For example, note that

〈[1]〉=A〈[0]〉+A−^1 〈[∞]〉,

hence

F([1]) =i

A−^1

A

=iA−^2 =i(i−^1 )=1.

To have the fraction value 1 for the tangle [1] is the reason that in the
definition ofF(T) we normalized byi. Statement 6 follows from the fact that
the bracket of the mirror image of a knotKis the same as the bracket of
K, but withAandA−^1 switched. For proving 7 we observe first that for any
2-tangleT,d(1/T)=n(T)andn(1/T)=d(T),where the overline denotes
the complex conjugate. Complex conjugates occur becauseA−^1 =Awhen
A=

√

i.Now,sinceF(T) is real, we have

F(T^1 )=id(T)/n(T)=−id(T)/n(T)= 1 /(in(T)/d(T)) = 1 /F(T)=
1 /F(T).

Statement 8 follows immediately from 6 and 7.This completes the proof.

For a related approach to the well definedness of the two-tangle frac-
tion, the reader should consult [29]. The double resmoothing idea originates
from [30].

Remark 2 For any knot or linkKwe define thedeterminantofKby the
formula

Det(K):=|〈K〉(

√

i)|,

where|z|denotes the modulus of the complex numberz.Thuswehavethe
formula

|F(T)|=

Det(N(T))

Det(D(T))

for any two-tangleT.
In other approaches to the theory of knots, the determinant of the knot is
actually the determinant of a certain matrix associated either to the diagram
for the knot or to a surface whose boundary is the knot. (See [7, 24] for
more information on these connections). Conway’s original definition of the
fraction [4] is∆N(T)(−1)/∆D(T)(−1) where∆K(−1) denotes the evaluation
of the Alexander polynomial of a knotKat the value− 1 .In fact,|∆K(−1)|=
Det(K),and with appropriate attention to signs, the Conway definition and
our definition using the bracket polynomial coincide for all two-tangles.