108 L.H. Kauffman and S. Lambropoulou
In this version of the tangle model for DNA recombination we have made
a blanket assumption that the substrate tangleSand the recombination tan-
gleRand all the tanglesS+nRwere rational. Actually, if we assume that
Sis rational and thatS+Ris rational, then it follows thatRis an integer
tangle. ThusSandRneccessarily form a DNA knitting machine under these
conditions. It is relatively natural to assume thatSis rational on the grounds
of simplicity. On the other hand it is not so obvious that the recombination
tangle should be an integer. The fact that the products of the DNA recom-
bination experiments yield rational knots and links, lends credence to the
hypothesis of rational tangles and hence integral recombination tangles. But
there certainly is a subtlety here, since we know that the numerator closure
of the sum of two rational tangles is always a rational knot or link. In fact, it
is here that some deeper topology shows that certain rational products from
a generalized knitting machine of the formKn=N(S+nR), whereSandR
are arbitrary tangles will force the rationality of the tanglesS+nR. We refer
the reader to [36, 40, 41] for details of this approach.
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