Topology in Molecular Biology

5 From Tangle Fractions to DNA 97

open

Fig. 5.24.A non-rational cut

by the Tait conjecture they will differ by flypes. We analyse all possible flypes
to prove that no new cases for study arise. Hence the proof becomes complete
at that point. We refer the reader to our study [17] for details.

Remark 3 The original proof of the classification of unoriented rational
knots by Schubert [14] proceeded by a different route than the proof we have
just sketched. Schubert used a two-bridge representation of rational knots
(representing the knots and links as diagrams in the plane with two special
overcrossing arcs, called the bridges). From the two-bridge representation,
one could extract a fractionp/q,and Schubert showed by means of a canon-
ical form, that if two such presentations are isotopic, then their fractions are
arithmetically equivalent (in the sense that we have described here). On the
other hand, Seifert [14] observed that the twofold branched covering space
of a two-bridge presentation with fractionp/qis a lens space of typeL(p, q).
Lens spaces are a particularly tractable set of three manifolds, and it is known
by work of Reidemeister and Franz [15, 34] thatL(p, q) is homeomorphic to
L(p′,q′) if and only ifp/qandp′/q′are arithmetically equivalent. Furthermore,
one knows that if knotsKandK′are isotopic, then their twofold branched
covering spaces are homeomorphic. Hence it follows that if two rational knots
are isotopic, then their fractions are arithmetically equivalent (via the re-
sult of Reidemeister and Franz classifying lens spaces). In this way Schubert
proved that two rational knots are isotopic if and only if their fractions are
arithmetically equivalent.

5.6 Rational Knots and Their Mirror Images

In this section we give an application of Theorem 2. An unoriented knot or
linkKis said to beachiralif it is topologically equivalent to its mirror image
−K. If a link is not equivalent to its mirror image then it is said bechiral.
One then can speak of thechiralityof a given knot or link, meaning whether
it is chiral or achiral. Chirality plays an important role in the applications of
knot theory to chemistry and molecular biology. It is interesting to use the