Knot Theory: Calculating the Alexander Polynomial

Question

I am going to be giving a talk about knot thoery in a few weeks and I will be discussing different knot invariants-one of which being the alexander polynomial

I am having a problem understanding how this linear equation is created using the method of going around anti-clockwise.

Can someone help me work through this?

Answer 1

The text is simply defining $c_i(r)$ to be that alternating sum with the terms being in the order given by going from the 'most clockwise' dot and going anti-clockwise around the crossing.

We start with $0$ then from the region with the first dot (with respect to looking anti-clockwise) we add the symbol corresponding to that region, as well as a multiplier of $t$ because that region has a dot - so we have $tr_j$.

We then subtract the next region which is anticlockwise from the first region, and also add the multiplier $t$ because this region has a dot. This gives us $tr_j-tr_k$ so far.

We carry on going round, this time adding the next symbol without a multiplier because there is no dot - giving $tr_j-tr_k+r_l$ and finally we subtract the final region around the circle, also with no multiplier because there is no dot, to give $$c_i(r)=tr_j-tr_k+r_l-r_m.$$ As to how this definition is useful, you'll have to carry on reading :).