Topology in Molecular Biology

36 A. Vologodskii

second contour ifi>M. All elements of the Alexander matrix exceptakk,

akk+1andakiare zero. The nonzero elements of thekth row are defined as

follows:

(1)k≤M, M > 1

(a) Fori=kori=k+1

akk=− 1 ,akk+1= 1 independent of the type of crossing;

(b) Fori=k,i=k+1,i≤M

akk=1,akk+1=−s;aki=s−1 for type I crossing,

akk=−s, akk+1=1;aki=s−1 for type II crossing;

(c) Fori>M

akk=1,akk+1=−t;aki=s−1 for type I crossing,

akk=−t, akk+1=1;aki=s−1 for type II crossing;

(2)k=M=1;i>M
akk=1−t, aki=s−1 independent of the type of crossing;
(3)k>M;N>M+1

(a) Fori=kori=k+1

akk=− 1 ,akk+1= 1 independent of the type of underpass;

(b) Fori=k,i=k+1,i>M

akk=1,akk+1=−t;aki=t−1 for type I crossing,

akk=−t, akk+1=1;aki=t−1 for type II crossing;

(c) Fori≤M

akk=1,akk+1=−s;aki=t−1 for type I crossing,

akk=−s, akk+1=1;aki=t−1 for type II crossing.

(4)k=N;N=M+1,i≤M
akk=1−s, aki=t−1 independent of the type of crossing.

These relationships hold under conditions that fork=Mone makes the

substitution (k+1)→1 and fork=Nthe substitution (k+1)→M+1. It

is also assumed that there is at least one crossing in each contour formed by

generators from different contours – in other case the contours are certainly

unlinked.

Then one has to calculate a minorAkjof orderN−1 of the Alexander

matrix and divide it by (s−1) ifj ≤M and by (t−1) ifj>M.The

resultant expression is multiplied by (±t−ms−n)(mandnare integers), so

that the polynomial so obtained has no negative powers, and the positive

powers are minimal, and the term with the largest total exponent must be

positive. The polynomial ∆(s, t) defined in this manner is calledthe Alexander

polynomial for the link of two contours. It is an invariant – it is rigorously

proved in the knot theory that the Alexander polynomials ∆(s, t) coincide for

equivalent links. For unlinked contours ∆(s, t)=0.

It is usually sufficient to calculate ∆(s, t) = 0 for the majority of problems

related with circular DNA molecules. The analysis shows that ∆(s, t)=0is

a more powerful topological invariant than the Gauss integral (which equals

∆(s, t) = 0) [26]. It has different values for the simplest links and for unlinked