knot theory - Difference between Alexander polynomial and Blanchfield pairing

The Blanchfield pairing has many formulations, I like to think of it as a sesquilinear form:

$$ A otimes A to Lambda / mathbb Z[t^pm] $$

where $A$ is the Alexander module and $Lambda$ is the field of fractions of $mathbb Z[t^pm]$. This pairing has to be a duality isomorphism, ie: the adjoint

$$ overline{A} to Hom_{mathbb Z[t^pm]} (A, Lambda/mathbb Z[t^pm]) $$

is an isomorphism of $mathbb Z[t^pm]$-modules. $overline{A}$ is $A$ but given the opposite action of $mathbb Z[t^pm]$ (you substitute $t longmapsto t^{-1}$ before multiplication by a polynomial)

The Blanchfield pairing can be anything of that form. So you take the Alexander module, and soup it up with such an isomorphism between $overline{A}$ and its ``Ext dual'' $Hom_{mathbb Z[t^pm]} (A, Lambda/mathbb Z[t^pm]) $. That is the extra information in the S-equivalence class.

edit: the pairing has a nice geometric interpretation. $A$ is $H_1(tilde C)$ where $tilde C to C$ is the universal abelian cover of the knot complement. Since $A$ is $mathbb Z[t^pm]$-torsion, given any $[x] in A$ let $p$ be such that $px = partial X$. Then you define the pairing $langle x, yrangle = (sum_i (X cap t^{i}y)t^i)/p$ provided $X$ and $y$ are transverse representatives when projected to $C$ (in any way that that makes sense). Here $cap$ is the standard algebraic intersection number of transverse chains.