5 From Tangle Fractions to DNA 995.7 The Oriented Case
Oriented rational knots and links arise as numerator closures of oriented ratio-
nal tangles. In order to compare oriented rational knots via rational tangles
we need to examine how rational tangles can be oriented. We orient ratio-
nal tangles by choosing an orientation for each strand of the tangle. Here we
are only interested in orientations that yield consistently oriented knots upon
taking the numerator closure. This means that the two top end arcs have to
be oriented one inward and the other outward. Same for the two bottom end
arcs. We shall say that two oriented rational tangles areisotopicif they are
isotopic as unoriented tangles, by an isotopy that carries the orientation of
one tangle to the orientation of the other. Note that, since the end arcs of
a tangle are fixed during a tangle isotopy, this means that the tangles must
have identical orientations at their four end arcsNW, NE, SW, SE. It follows
that if we change the orientation of one or both strands of an oriented rational
tangle we will always obtain a non-isotopic oriented rational tangle.
Reversing the orientation of one strand of an oriented rational tangle may
or may not give rise to isotopic oriented rational knots. Figure 5.26 illustrates
an example of non-isotopic oriented rational knots, which are isotopic as un-
oriented knots.
Reversing the orientation of both strands of an oriented rational tangle will
always give rise to two isotopic oriented rational knots or links. We can see
this by doing a vertical flip, as Fig. 5.27 demonstrates. Using this observation
we conclude that, as far as the study of oriented rational knots is concerned,
all oriented rational tangles may be assumed to have the same orientation for
~
close closeFig. 5.26.Non-isotopic oriented rational Links,
180 o~
Fig. 5.27.Isotopic oriented rational knots and links