Quandle Polynomial 2-Cocycle Knot Invariants for Alexander Quandles

Polynomial cocycles for Alexander quandles were used for these calculations. The polynomial f(x,y)=(x-y)^(p^m) is a 2-cocycle for any Alexander quandle with Z_p coefficients (p denotes a prime), and f(x,y)=(x-y)^(p^m) y^(p^n) is a 2-cocycle for Z_p Alexander quandles modulo g(t) if g(t) divides (t^(p^m+p^n)-1).

We computed the invariants for knots in the table up to 9 crossings. We computed only those knots whose Alexander polynomials are not coprime mod p with g(t).(By Inoue's theorem only such knots are colored non-trivially.) Here the Alexander quandle we use is Z_p coefficients mod g(t).

When we found non-trivial values in our calculations, it is marked ``non-trivial.'' Note, however, that there are non-trivial (i.e., non-coboundary) cocycles that give trivial invariant values. It may be possible, for example, that it takes non-trivial values only for ``virtual knots'' that are non-classical.


First we try cocycles f(x,y)=(x-y)^(p^m) for quandles in Sam Nelson's table [Nel03].

Comments: It turned out that all of the following had trivial values (i.e., positive integers that represent the number of colorings). It may be conjectured that this cocycle is a coboundary for any Alexander quandle.


Next we try cocycles of the form f(x,y)=(x-y)^(p^m) y^(p^n).

For Z_2 coefficients:


For Z_3 coefficients: