104 L.H. Kauffman and S. Lambropoulou
L = N([[3], [1], [1], [1], [3]])
Fig. 5.31.An example of a strongly invertible linkN ( )
[-3]
_^1 N ( + [0])
[-3]_^1 N ( + [1])
[-3]_^1
Fig. 5.32.Global picture of recombinationmolecule or by integrating a block of alien DNA into a host genome. For a
closed molecule of DNA a global picture of the recombination would be as
shown in Fig. 5.32, where double-stranded DNA is represented by a single
line and the recombination sites are marked with points. This picture can be
interpreted asN(S+ [0])−→N(S+ [1]),forS=1/[−3] in this example. This
operation can be repeated as in Fig. 5.33. Note that the [0]−[∞] interchange
of Fig. 5.10 can be seen as the first step of the process.
In this depiction of recombination, we have shown a local replacement of
the tangle [0] by the tangle [1] connoting a new cross-connection of the DNA
strands. In general, it is not known without corroborating evidence just what
the topological geometry of the recombination replacement will be. Even in
the case of a single half-twist replacement such as [1], it is certainly not obvious
beforehand that the replacement will always be [+1] and not sometimes the
reverse twist of [−1]. It was at the juncture raised by this question that a
combination of topological methods in biology and a tangle model using knot
theory developed by C. Ernst and D.W. Sumners resolved the issue in some
specific cases. (See [36, 37] and references therein.)
On the biological side, methods of protein coating developed by N. Coz-
zarelli, S.J. Spengler and A. Stasiak et al. in [38] made it possible for the first
time to see knotted DNA in an electron micrograph with sufficient resolution