I realize the poster is probably long gone by now, but I am in a knot theory class currently and might as well post some content in this thread. I can try to give more knot theory formulas if there is any request (it doesn't seem like a hugely popular field, however).
The Conway Polynomial
The Conway polynomial is a polynomial invariant of knots and links described by the following three axioms:
Axiom 1: For each oriented knot or link K there is an associated polynomial ∇K(z) ∈ Z[z] (Z[z] is the ring of polynomials in z with integer coefficients). If one knot K is ambient isotopic to another knot K' ( K ~ K'), then ∇K = ∇K'.
Axiom 2: If K is ambient isotopic to the unknot (K ~ O), then ∇K = 1.
Axiom 3: Suppose that three knots or links K+, K-, and L differ at one crossing in the manner shown below:
Then ∇K+ - ∇K- = z∇L.
Axiom 1 tells us that for any knot or link there exists a Conway polynomial; Axioms 2 and 3 give us a way to find it. Tomorrow I shall post an example of how to use these axioms to find the Conway polynomial of a knot (using a specific example, most likely the trefoil, but maybe some others), and perhaps I shall also describe the Jones polynomial, the HOMFLY polynomial, the chromatic polynomial, and more.
Edit: Wow, sorry about that, I could have sworn this topic had been replied to very recently, and I didn't realize it had been moved from the formulas section.