5 From Tangle Fractions to DNA 101[0] =
[∞]=><
[+1] =χ.
Note that connectivity type [0] yields two-component rational links, while
type [+1] or [∞] yields one-component rational links. Also, adding a bottom
twist to a rational tangle of connectivity type [0] will not change the connec-
tivity type of the tangle, while adding a bottom twist to a rational tangle of
connectivity type [∞] will switch the connectivity type to [+1] and vice versa.
While the connectivity type of unoriented rational tangles may be [0], [+1]
or [∞],note that an oriented rational tangle of type I will have connectivity
type [0] or [∞] and an oriented rational tangle of type II will have connectivity
type [0] or [+1].
Further, we need to keep an accounting of the connectivity of rational
tangles in relation to the parity of the numerators and denominators of their
fractions. We refer the reader to our Study [17] for a full account.
We adopt the following notation:estands forevenandostands forodd.
Theparityof a fractionp/qis defined to be the ratio of the parities (eoro)
of its numerator and denominatorpandq. Thus the fraction 2/3 is of parity
e/o.The tangle [0] has fraction 0 = 0/ 1 ,thus paritye/o; the tangle [∞]has
fraction∞=1/ 0 ,thus parityo/e; and the tangle [+1] has fraction 1 = 1/ 1 ,
thus parityo/o.We then have the following result.
Theorem 5. A rational tangleThas connectivity typeif and only if its
fraction has paritye/o.Thas connectivity type><if and only if its fraction
has parityo/e.Thas connectivity typeχif and only if its fraction has parity
o/o. (Note that the formal fraction of[∞]itself is 1 / 0 .) Thus the linkN(T)
has two components if and only ifThas fractionF(T)of paritye/o.
We will now proceed with sketching the proof of Theorem 3. We shall
prove Schubert’s oriented theorem by referring to our previous work on the
unoriented case and then analyzing how orientations and fractions are related.
Our strategy is as follows: Consider an oriented rational knot or link diagram
Kin the formN(T), whereTis a rational tangle in continued fraction form.
Then any other rational tangle that closes to this knotN(T) is available, up
to bottom twists if necessary, as a cut from the given diagram. If two rational
tangles close to giveK as an unoriented rational knot or link, then there
are orientations on these tangles, induced fromKso that the oriented tangles
close to giveKas an oriented knot or link. The two tangles may or may not be
compatible. Thus, we must analyze when, comparing with the standard cut for
the rational knot or link, another cut produces a compatible or incompatible
rational tangle. However, assuming the top orientations are the same, we
can replace one of the two incompatible tangles by the tangle obtained by
adding a twist at the bottom.It is this possible twist difference that gives rise