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:

K_{+}

K_{-}

L

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.

*Last edited by Zhylliolom (2007-02-21 19:45:06)*