78 L.H. Kauffman and S. Lambropoulou
Definition 2 Acontinued fraction in integer tanglesis an algebraic descrip-
tion of a rational tangle via a continued fraction built from the tangles [a 1 ],
[a 2 ],...,[an] with all numerators equal to 1, namely an expression of the type:
[[a 1 ],[a 2 ],...,[an]] := [a 1 ]+1
[a 2 ]+···+[an− 11 ]+ 1
[an]fora 2 ,...,an∈Z−{ 0 }andneven or odd. We allow that the terma 1 may
be zero, and in this case the tangle [0] may be omitted. A rational tangle
described via a continued fraction in integer tangles is said to be incontinued
fraction form.Thelengthof the continued fraction is arbitrary – in the pre-
vious formula illustrated with lengthn– whether the first summand is the
tangle [0] or not.
It follows from Lemma 3.2 that inductivelyevery rational tangle can be
written in continued fraction form.Lemma 3.2 makes it easy to write out the
continued fraction form of a given rational tangle, since horizontal twists are
integer additions, and multiplications by vertical twists are the reciprocals of
integer additions. For example, Fig. 5.17 illustrates the rational tangle
[2] +
1
[−2] +[3]^1
,
Fig. 5.17 illustrates the rational tangle
[2] +
1
[2] +[3]^1
.
Note that (
[c]∗1
[b])
+[a]has the continued fraction form
[a]+1
[b]+[^1 c]=[[a],[b],[c]].ForT=[[a 1 ],[a 2 ],...,[an]] the following statements are now straightforward.
1 .T+[±1] = [[a 1 ±1],[a 2 ],...,[an]],
2. T^1 = [[0],[a 1 ],[a 2 ],...,[an]],
3. −T=[[−a 1 ],[−a 2 ],...,[−an]].We now recall some facts about continued fractions. (See for example
[21–24]). In this chapter we shall only consider continued fractions of the
type
[a 1 ,a 2 ,...,an]:=a 1 +1
a 2 +···+an− 11 + 1
an