I understand that there are multiple ways to calculate the knot determinant, one is through the Alexander polynomial, the other is by creating another matrix which uses the linesections and crossings, the last method is explained in this pdf.
My problem lies in the fact that I don't understand how both can give the same number, so in other words, how can two different things calculate the same thing. I think it's probably because of some deeper underlying thing which can be approached using these different methods, but I don't know what this 'thing' is.
So my question is, what is the connection between these two methods of calculating the knot determinant?
A useful answer would be a proof (or a referral to one) of the fact that these two methods will always give the same number or some insights in helping me understand why they must be the same.
Edit: My definition of the Alexander polynomial is the polynomial which you get after computing the determinant of the Alexander matrix, you can find a more detailed explanation in this Wikipedia article