Second Alexander Polynomial

Alexander's original paper in which he defined what is now called the Alexander polynomial included a definition of a sequence of polynomial invariants, each dividing the next. They are most easily defined over the rational numbers. The rational homology of the infinite cyclic cover a knot K is a finitely generated torsion module over the Laurent polynomial ring Q[t,t^{-1}]. This is the rational Alexander module. It can written as finite direct sum of cyclic modules of the form Q[t,t^{-1}]/< f _i (t) >, where each successive f _i divides the previous one.

The n^th Alexander polynomial is the product of the f _i, starting with the n^th one. This is well-defined up to multiplication by a unit in Q[t,t^{-1}], and thus a representative is always chosen to be a primitive interger polynomial. It can be proved they are symmetric and evaluate to be ±1 at 1.

As an example, for the knot 9₄₀, the Alexander module is Q[t,t^{-1}]/< (t^2-1+t)(t^2-3t+1) > + Q[t,t^{-1}]/<(t^2-3t+1) >. Thus, the first (classical) Alexander polynomial is (t^2-1+t)(t^2-3t+1)^2 and the second is (t^2-3t+1).

The third Alexander polynomial is the trivial polynomial f₃(t) = 1 for every prime knot with 13 crossings or less.

The general theory is presented in many basic references in knot theory, including [1, 2]. Often these polynomials are defined in terms of a nested sequence of Alexander ideals in the ring Z[t,t^{-1}]. Each Alexander ideal is contained in a unique minimal principal ideal; these are generated by the Alexander polynomials.

Computations were carried out using algorithms described in [3].