76 L.H. Kauffman and S. Lambropoulou
T
T + = T
T =
Fig. 5.8.Creating new rational tanglesT
N
N(T)
T
D
D(T)
T ~ T
Fig. 5.9.The numerator and denominator of a two-tangletangle. Using flypes and flips inductively on subtangles one can always bring
the twists to the right or bottom of the rational tangle. We shall then say
that the rational tangle is instandard form. Thus a rational tangle in stan-
dard form is created by consecutive additions of the tangles [±1]only on
the rightand multiplications by the tangles [±1]only at the bottom,start-
ing from the tangles [0] or [∞].For example, Fig. 5.1 illustrates the tangle
(([3]∗ 1 /[−2]) + [2]), while Fig. 5.17 illustrates the tangle (([3]∗ 1 /[2]) + [2]) in
standard form. Figure 5.8 illustrates addition on the right and multiplication
on the bottom by elementary tangles.
We also have the followingclosingoperations, which yield two different
knots: theNumeratorof a two-tangleT, denoted byN(T), obtained by joining
with simple arcs the two upper endpoints and the two lower endpoints ofT,
and theDenominatorof a two-tangleT, obtained by joining with simple arcs
each pair of the corresponding top and bottom endpoints ofT, denoted by
D(T) (Fig. 5.9). We haveN(T)=D(Tr)andD(T)=N(Tr). We note that
every knot or link can be regarded as the numerator closure of a two-tangle.
We obtainD(T)fromN(T) by a [0]–[∞] interchange, as shown in Fig. 5.10.
This “transmutation” of the numerator to the denominator is a precursor to
the tangle model of a recombination event in DNA, see Sect. 5.9. The [0]–[∞]
interchange can be described algebraically by the equations: