34 A. Vologodskii
Fig. 3.9.The simplestknotsand their Alexander polynomials, ∆(t). All four knots
that can be drawn with less than six intersections are shown. For an unknotted
contour ∆(t)=1
Fig. 3.10.On the calculation of an Alexander polynomial forknots.Herex 1 ,x 2 ,x 3 ,
andx 4 are the generators, and 1, 2, 3, and 4 are the crossing points in the projection
of the knot
Fig. 3.11.The two types of crossingsgenerators and renumber them, having selected arbitrarily the first generator.
The crossing that separates thekth and (k+ 1)th generators will be called the
kth crossing. The crossings are of two types (Fig. 3.11). Thus each crossing
is characterized by its number, by its type (I or II), and by the number of
generator passing over it.
Now the knot can be correlated with a square Alexander matrix, in which
thekth row corresponds to thekth crossing and which consists ofNelements
(Nis the total number of crossings in the projection of the knot). Here all
the elements exceptakk,akk+1andaki(iis the number of the overpassing
generators) are zero. The nonzero elements of thekth row are determined as
follows: