Polynomial (Mochizuki) cocycles

We consider Alexander quandles and their quandle cocycles with the coefficient group of the form $X=\mathbb{Z}_p[t,t^{-1}]/h(t)=A$. In this case some $2$- and $3$-cocycles are constructed by polynomial functions. These constructions were given in [Mochi03].

In general, functions of the form

$$f(x_1, \ldots, x_n)=(x_1-x_2)^{p^{m_1}} \cdots (x_{n-1}-x_n)^{p^{m_{n-1}}} x_n^{a_n}$$

are $n$-cocycles for a variety of Alexander quandles mod $p$, where $p$ is a prime and $a_n$ is either 0 or a power of $p$. In our case, we used the following cocycles.

• $f(x,y)=(x-y)^{p^{n}}$ is a $2$-cocycle for any Alexander quandle mod $p$.

• $f(x,y)=(x-y)^{p^{m_1}}y^{p^{m_2}}$ is a $2$-cocycle for an Alexander quandle $\mathbb{Z}_p[t, t^{-1}]/g(t)$ if $g(t)$ divides $(t^{p^{m_1}+p^{m_2}}-1)$.

• $f(x,y, z)=(x-y)^{p^{m_1}} (y-z)^{p^{m_2}}$ is a $3$-cocycle for any Alexander quandle mod $p$.

• $f(x,y, z)=(x-y)^{p^{m_1}} (y-z)^{p^{m_2}} z^{p^{m_3}}$ is a $3$-cocycle for an Alexander quandle $\mathbb{Z}_p[t, t^{-1}]/g(t)$ if $g(t)$ divides $(t^{p^{m_1}+p^{m_2}+p^{m_3}}-1)$.

There is another type of $3$-cocycles Mochizuki constructed for dihedral quandles $R_p$ for prime $p$. It is given by the formula

$$\theta (x,y,z)= (x-y) [(2 z^p - y^p) - (2z-y)^p ]/p$$   mod$$\; p,$$

where the numerator computed in $\mathbb{Z}$ is divisible by $p$, and then after dividing it by $p$, the value is taken as an integer modulo $p$.