Quandle Polynomial 3-Cocycle Knot Invariants for Alexander Quandles

Polynomial cocycles for Alexander quandles were used for these calculations. The functions f(x,y,z)=(x-y)^(p^m) (y-z)^(p^n) are 3-cocycles for any Alexander quandles, and we tried some of these cocycles.

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).

• Quandle: Z_2 mod(t^2+t+1), 3-cocycle: f(x,y,z) = (x-y) (y-z)^2 (non-trivial)

Comments: All non-trivial values had the form of k*[16+48*u^t], so we conjecture that this is always the case.

• Quandle: Z_2 mod(t^3+1), 3-cocycle: f(x,y,z) = (x-y) (y-z)^2 (non-trivial)

Comments: All non-trivial values had the form of k*[64+192*u^(t^2+1)], so we conjecture that this is always the case.

• Quandle: Z_2 mod(t^3+t+1), 3-cocycle: f(x,y,z) = (x-y) (y-z)^2
• Quandle: Z_2 mod(t^3+t^2+1), 3-cocycle: f(x,y,z) = (x-y) (y-z)^2
• Quandle: Z_2 mod(t^3+t^2+t+1), 3-cocycle: f(x,y,z) = (x-y) (y-z)^2

Comments: For these, we did not find non-trivial values up to 9 crossings, although there may be non-trivial colorings.

• Quandle: Z_3 mod(t^2+1), 3-cocycle: f(x,y,z) = (x-y) (y-z)^3 (non-trivial)

Comments: There were two non-trivial values, 81+324*u^(2*t+1)+324*u^(t+2) for all except 9_40, which had 2673+1944*u^(t+2)+1944*u^(2*t+1). We note that the latter is not a multiple of the former.

• Quandle: Z_3 mod(t^2+2), 3-cocycle: f(x,y,z) = (x-y) (y-z)^3
• Quandle: Z_3 mod(t^2+t+1), 3-cocycle: f(x,y,z) = (x-y) (y-z)^3
• Quandle: Z_3 mod(t^2+t+2), 3-cocycle: f(x,y,z) = (x-y) (y-z)^3

Comments: These were trivial up to 9 crossings.

• Quandle: Z_3 mod(t^2+2t+1), 3-cocycle: f(x,y,z) = (x-y) (y-z)^3 (non-trivial)

Comments: There were three non-trivial values up to 9 crossings, 243+486*u^(t+1), 243+486*u^(2*t+2), and 729+2916*u^(t+1)+2916*u^(2*t+2).

• Quandle: Z_3 mod(t^2+2t+2), 3-cocycle: f(x,y,z) = (x-y) (y-z)^3

Comments: This was trivial up to 9 crossings.

• Quandle: Z_5 mod(t^2-t+1), 3-cocycle: f(x,y,z) = (x-y) (y-z)^5 (non-trivial)
• Quandle: Z_7 mod(t^2-t+1), 3-cocycle: f(x,y,z) = (x-y) (y-z)^7
Comments: I had to skip knot 8_18. This has to many colorings for now. I will have to adjust the program for this one.