The Alexander-Conway Polynomial

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The Alexander polynomial $A(K)$ and the Conway polynomial $C(K)$ of a knot $K$ always satisfy $A(K)(t)=C(K)({\sqrt {t}}-1/{\sqrt {t}})$. Let us verify this relation for the knot 8_18:

The determinant of a knot $K$ is $|A(K)(-1)|$. Hence for 8_18 it is

Alternatively (see The Determinant and the Signature):

$V_{2}(K)$, the (standardly normalized) type 2 Vassiliev invariant of a knot $K$ is the coefficient of $z^{2}$ in its Conway polynomial:

Alternatively (see Finite Type (Vassiliev) Invariants),

Sometimes two knots have the same Alexander polynomial but different Alexander ideals. An example is the pair K11a99 and K11a277. They have the same Alexander polynomial, but the second Alexander ideal of the first knot is the whole ring ${\mathbb {Z} }[t]$ while the second Alexander ideal of the second knot is the smaller ideal generated by $3$ and by $1+t$:

Finally, the Alexander polynomial attains values on the knots known to KnotTheory`: