98 L.H. Kauffman and S. Lambropoulou
classification of rational knots and links to determine their chirality. Indeed,
we have the following well-known result (for example see [5] and also page 24,
Exercise 2.1.4 in [9]):
Theorem 4.LetK=N(T)be an unoriented rational knot or link, presented
as the numerator of a rational tangleT.Suppose thatF(T)=p/qwithpand
qrelatively prime. ThenKis achiral if and only ifq^2 ≡−1 (modp).It follows
that achiral rational knots and links are all numerators of rational tangles of
the form[[a 1 ],[a 2 ],...,[ak],[ak],...,[a 2 ],[a 1 ]]for any integersa 1 ,...,ak.
Note that in this description we are using a representation of the tangle
with an even number of terms. The leftmost twists [a 1 ] are horizontal, thus
the rightmost starting twists [a 1 ] are vertical.
Proof.With−Tthe mirror image of the tangleT, we have that−K=N(−T)
andF(−T)=p/(−q).IfK is topologically equivalent to−K, thenN(T)
andN(−T) are equivalent, and it follows from the classification theorem for
rational knots that eitherq(−q)≡1 (modp)orq≡−q(modp).Without loss
of generality we can assume that 0<q<p.Hence 2qis not divisible byp
and therefore it is not the case thatq≡−q(modp).Henceq^2 ≡−1 (modp).
Conversely, ifq^2 ≡−1 (modp),then it follows from the palindrome the-
orem (described in the previous section) [17] thatthe continued fraction ex-
pansion ofp/qhas to be symmetric with an even number of terms.It is then
easy to see that the corresponding rational knot or link, sayK=N(T),is
equivalent to its mirror image. One rotatesKby 180◦in the plane and swings
an arc, as Fig. 5.25 illustrates. This completes the proof.
In [35] the authors find an explicit formula for the number of achiral
rational knots among all rational knots withncrossings.
180 rotationK
swing arcFig. 5.25.An achiral rational link