Topology in Molecular Biology

96 L.H. Kauffman and S. Lambropoulou

special cut

~

~ = [-1] -

open

to obtain

_^1

T

T

K = N([1] +T) = T

T

T

Fig. 5.22.A special cut

1

(^23454321)
1
2 3 4 5 4 3 2
1
112233 55 6 67 7
Standard Cuts Palindrome Cuts
Special Cuts
44
112233 5566 7
44 7
Fig. 5.23.Standard, palindrome and special cuts
the cases that have been already verified. It is convenient to say that reduced
fractionsp/qandp′/q′arearithmetically equivalent, writtenp/q∼p′/q′if
p=p′and eitherqq′≡1 (modp)orq≡q′(modp). In this language,
Schubert’s theorem states that two rational tangles close to form isotopic
knots if and only if their fractions are arithmetically equivalent.
In Fig. 5.24 we illustrate one example of a cut that is not allowed since it
opens the knot to a non-rational tangle.
In the second stage of the proof we want to check the arithmetic equiv-
alence for two different given knot diagrams, numerators of some rational
tangles. By Lemma 2 the two knot diagrams may be assumed alternating, so