The vector notation for a polynomial consists of a sequence of integers: the first two are the lowest and highest degrees of the polynomial. The remaining terms are the coefficients.
As an example: the polynomial 2-3*t+ 2*t^2 is represented by {0, 2, 2, -3, 2}.
This can be iterated: for the HOMFLY polynomial of 6_3 we have the polynomial (-v^(-2)+ 3-v^2)+ (-v^(-2)+ 3-v^2)*z^2+ (1)*z^4 is represented by {0, 2, {-1, 1, -1, 3, -1}, {-1, 1, -1, 3, -1}, {0, 0, 1}}. Notice that since this polynomial is even (in z) the highest power is divided by 2 so that the 0 coefficients can be skipped.