114 C. Cerf and A. Stasiak
If one considers all alternating knots and links, there is no simple function
that can relate the writhe to the minimal crossing number. However, Huang
and Lai [11] used (6.1) to calculate the writhe of ideal knots belonging to
selected families of Conway knots, and observed that within these families the
writhe can be expressed as a linear function of the minimal crossing number.
We show here how to express the writhe as a linear function of the minimal
crossing number in any individual Conway family of rational knots and link
(Fig. 6.1). A related publication has been submitted to theNew Journal of
Physics.
6.2 Rational Tangles and Rational Links
Rational tangles and rational links have been introduced by Conway in 1970
[12]. Arational tangleis a region of a knot or link projection composed of a
succession of vertical and horizontal rows of crossings, and is denoted by a
sequence of numbers corresponding to the number of crossings in each row
(see Fig. 6.2a). To avoid confusion, one always has to end with a horizontal
row. If the latter contains no crossing, the sequence will end with (0). All
crossings are done in order for the projection to be alternating (each strand
alternatively goes over and under other strands). Figure 6.2a a shows positive
rational tangles. If each crossing is inversed (i.e., the mirror image is consid-
ered), one gets negative rational tangles that will be denoted by a sequence of
negative numbers. Conway proved that a rational tangle denoted by a mixed
sequence of positive and negative numbers (i.e., a nonalternating projection)
is always topologically equivalent to an alternating projection with all posi-
tive or all negative numbers. We will go on with positive rational tangles only.
Extension to negative rational tangles is obvious.
Theclosureof a rational tangle is the operation of rejoining the two upper
free ends and the two lower free ends on the projection (see Fig. 6.2b). The
link we obtain is called arational link(also calledtwo-bridge linkorfour-
plat). Rational links are completely classified. They all have either one or
two components (let us recall that a one-component link is a knot). Nearly
all knots and links naturally occurring in closed polymer chains are rational
links.
6.3 Writhe of Families of Rational Links
We now examine some families of rational links and calculate theirPWr.
6.3.1 Tangles with One Row, Denoted by (a),aPositive Integer
Figure 6.3a shows an example of such a tangle, witha= 5. Since there is only
one row, the minimum crossing numbern=a(5 in this case).