Topology in Molecular Biology

5 From Tangle Fractions to DNA 103

5.8 Strongly Invertible Links.....................................

An oriented knot or link is invertible if it is oriented isotopic to the link
obtained from it by reversing the orientation of each component. We have
seen (see Fig. 5.27) that rational knots and links are invertible. A linkLof
two components is said to bestrongly invertibleifLis ambient isotopic to
itself with the orientation of only one component reversed. In Fig. 5.30 we
illustrate the linkL=N([[2],[1],[2]]).This is a strongly invertible link as is
apparent by a 180◦vertical rotation. This link is well known as the Whitehead
link, a link with linking number zero. Note that since [[2],[1],[2]] has fraction
equal to 1 + 1/(1 + 1/2) = 8/3 this link is non-trivial via the classification of
rational knots and links. Note also that 3·3=1+1· 8.
In general we have the following. For our proof, see [17].

Theorem 6.LetL=N(T)be an oriented rational link with associated tangle
fractionF(T)=p/qof paritye/o,withpandq relatively prime and|p|>
|q|.ThenLis strongly invertible if and only ifq^2 =1+upwithuan odd
integer. It follows that strongly invertible links are all numerators of rational
tangles of the form[[a 1 ],[a 2 ],...,[ak],[α],[ak],...,[a 2 ],[a 1 ]]for any integers
a 1 ,...,ak,α.

(See Fig. 5.31 for another example of a strongly invertible link.) In this
case the link isL=N([[3],[1],[1],[1],[3]]) withF(L)=40/ 11 .Note that
112 =1+3· 40 ,fitting the conclusion of Theorem 6.

5.9 Applications to the Topology of DNA

DNA supercoils, replicates and recombines with the help of certain enzymes.
Site-specific recombinationis one of the ways nature alters the genetic code
of an organism, either by moving a block of DNA to another position on the

N([[2], [1], [2]]) = W

the Whitehead Link

F(W) = 2+1/(1+1/2) = 8/3

3 3 = 1 + 1 8..

Fig. 5.30.The whitehead link is strongly invertible