RE: The Intellectual Thread
03-13-2018, 03:35 AM
Let K⊂S3 be an oriented knot or link, and F⊂S3 a connected oriented spanning surface for K. Let θ:H1(F)⨯H1→Z be the Seifert paring.
Two polynomials f(t),g(t)∈Z[t] are said to be balanced (written f≐g) if there is a non-negative integer n such that ±tnf(t)=g(t) or ±tng(t)=f(t). This definition is also extended to rational functions. Thus t+1/t and t2+1 are balanced and we write t+1/t≐t2+1.
Let K, F, θ be as above. The Alexander polynomial AK(t) is the balance class of the polynomial AK(t) = D(θ-tθ'). (It follows from S-equivalence that this determinant is well-defined on isotopy classes of knots and links up to multiplication by factors of the form ±tn)
Two polynomials f(t),g(t)∈Z[t] are said to be balanced (written f≐g) if there is a non-negative integer n such that ±tnf(t)=g(t) or ±tng(t)=f(t). This definition is also extended to rational functions. Thus t+1/t and t2+1 are balanced and we write t+1/t≐t2+1.
Let K, F, θ be as above. The Alexander polynomial AK(t) is the balance class of the polynomial AK(t) = D(θ-tθ'). (It follows from S-equivalence that this determinant is well-defined on isotopy classes of knots and links up to multiplication by factors of the form ±tn)
![[Image: nifOFwR.png]](https://i.imgur.com/nifOFwR.png)
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