3 Monte Carlo Simulation of DNA Topological Properties 35(1) Wheni=kori=k+ 1, thenak,k=−1andak,k+1= 1, independent of
the type of crossings
(2) Wheni=kandi=k+ 1, then
akk=1,ak,k+1=−tandakk=t−1 for a type I crossing,
akk=−t, ak,k+1=1andakk=t−1foratypeIIcrossing.
These relationships hold under condition that fork=None makes the
substitutionk+1→1.
To discriminate closed chains into two categories, knotted versus unknot-
ted, it is sufficient, in the most cases, to calculate ∆(t) at one pointt=−1.
The values oft=−1andt=−1 distinguish the great majority of all 166
knots that can be drawn with less than 11 intersections on their projection [42],
However, there are topologically different knots that have the same Alexan-
der polynomials. In particular,t=−1 of a knot and its mirror image are
identical, although very often such knots are topologically different. Thus,
other methods are required if we want to distinguish among such knots. The
problem can be solved by calculations of more powerful invariants, like the
Jones polynomial [43], although this requires much more computer time [44].
In some cases calculation of Wr helps to distinguish between a knot and its
mirror image [45].3.5.2 Links
To define the topology of two chains, one can calculate the Alexander poly-
nomial for two curves, Wr...[46]. The Alexander matrix for two chains is
constructed similar to the matrix for one chain. Two contours are projected
on an arbitrary chosen plane (see Fig. 3.12).
Renumbering of the generators,xk, and the corresponding crossings starts
in one contour and continues in the other. We denote the number of crossings
on the first contour byM. Renumbering of the generators and crossings in
the second contour starts fromM+ 1 and ends atN. Thus, the overpassing
generatorxifor crossingkbelongs to the first contour ifi≤Mand to theFig. 3.12.Calculation of an Alexander polynomial for links. Theprojectionis shown
with all generators and crossing points