**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 ∇[sub]K[/sub](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 ∇[sub]K[/sub] = ∇[sub]K'[/sub].

Axiom 2: If K is ambient isotopic to the unknot (K ~ O), then ∇[sub]K[/sub] = 1.

Axiom 3: Suppose that three knots or links K[sub]+[/sub], K[sub]-[/sub], and L differ at one crossing in the manner shown below:

K[sub]+[/sub]

K[sub]-[/sub]

L

Then ∇[sub]K+[/sub] - ∇[sub]K-[/sub] = z∇[sub]L[/sub].

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.

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