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.

hi,
How are you ?
I want any information , any topic ,website or any definition can help me to understand and explain these searching ( THE CONWAY POLYNOMIAL for louis H.Kauffman )because I do not understand it

thank you for your help

Moved to Help Me! - Ricky