Topology in Molecular Biology

94 L.H. Kauffman and S. Lambropoulou

and

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

1

[3] +[2]^1

both close to the same rational knot, shown at the bottom of the figure. The
two tangles are different, since they have different corresponding fractions:

F(T)=2+

1

3+^14

=

30

13

and F(S)=4+

1

3+^12

=

30

7

.

Note that the product of 7 and 13 is congruent to 1 modulo 30.

More generally, consider the following two fractions:

F=[a, b, c]=a+

1

b+^1 c

and G=[c, b, a]=c+

1

b+^1 a

.

We find that

F=a+c

1

cb+1

=

abc+a+c

bc+1

=

P

Q

,

while

G=c+a

1

ab+1

=

abc+c+a

ab+1

=

P

Q′

.

Thus we found thatF=P/QandG=P/Q′,where

QQ′=(bc+ 1)(ab+1)=ab^2 c+ab+bc+1=bP+1.

Assuming thata,bandcare integers, we conclude that

QQ′≡1 (modP).

This pattern generalizes to arbitrary continued fractions and their palin-
dromes (obtained by reversing the order of the terms), i.e.If{a 1 ,a 2 ,...,an}is
a collection of n non-zero integers, and ifA=[a 1 ,a 2 ,...,an]=P/Q and B=
[an,an− 1 ,...,a 1 ]=P′/Q′, then P=P′andQQ′≡(−1)n+1(modP).We will
be referring to this as the palindrome theorem. The palindrome theorem is a
known result about continued fractions. (For example, see [5] and [17]). Note
that we neednto be odd in the previous congruence. This agrees with Re-
mark 3.1 that without loss of generality the terms in the continued fraction
of a rational tangle may be assumed to be odd.
Finally, Fig. 5.20 illustrates another basic example for the unoriented Schu-
bert theorem. The two tanglesR= [1]+1/[2] andS=[−3] are non-isotopic by
the Conway theorem, sinceF(R)=1+1/2=3/2 whileF(S)=−3=3/− 1.
But they have isotopic numerators:N(R)∼N(S),the left-handed trefoil.
Now2iscongruentto−1modulo3,confirming Theorem 2.
We now analyse the above example in general. From the analysis of the
bottom twists we can assume without loss of generality that a rational tangleR