6 Writhe Versus the Number of Crossings 115aa (odd)+1 crossingsnullificationa (even)+1 crossingsnullificationa (even)-1 crossingsnullificationa (even)a
bc
de
Fig. 6.2.(a) A rational tangle with one row ofacrossings. (b)Ifais odd, the
closure of the tangle gives rise to a knot, whose nullification is shown. (c)Ifais
even, the closure of the tangle gives rise to a two-component link. Depending on the
orientation chosen for the second component, two different situations occur, shown
in (d)and(e)
a Odd
The rational knot obtained by the closure of such a tangle is the family
containing knots 3 1 , 51 , 71 , etc. Figure. 6.3b shows the nullification process [9]
applied to those knots. Crossings are successively nullified (or smoothed) until
the unknot is reached, forbidding at each step the apparition of a disconnected
component. The sum of the signs of the nullified crossings iswx,thesumof
the signs of the remaining crossings iswy. In this case, nullifyinga−1 positive