Alternative calculation of
The alternative calculation is very efficient. I cannot prove
that it delivers the goods without recourse to algebraic
Corollary , 9.5 for the proof.
Choose a chess boarding, of the knot diagram. Define a
(symmetric) matrix as follows. Both rows and columns are labeled by the
white regions. Attach to each crossing an index , as shown.
The matrix entry for row A and column B. With A not equal to B is
the sum of indices where A meets B. The diagonal entries are chosen so
that row and column sums are zero.
This is the Goeritz matrix (except that I changed all signs) of the
Delete a row and column and compute the determinant.
Example The pretzel link P(p,q,r)
There are only three white regions labelled A B C
and p + q - 1 black regions.
Now show that the determinant of 10124
is 1 and hence cannot be coloured mod any number. Here
another diagram for 10124
is another pretzel knot with determinant 9 which comes up in lecture 9