Recently I learned the Alexander Polynomial of a knot and how to find the polynomial for a given knot. Now there are some questions arise, I am trying to give some classification of hyperbolic knots, basically, I want to classify the hyperbolic knot groups, for the knots in $S^3$. Also, I want to find finite index subgroups of hyperbolic knot groups, which is why I have the following questions.
- For a given Alexander Polynomial can we classify all the knots?
- Suppose I want to find all knots whose Alexander Polynomial polynomial is = 1.
Thanks in advance