Polynomial Quandle Cocycles

To actually compute the quandle cocycle invariants from the definition, we need to have an explicit cocycles. In this section, we present quandle cocycles of Alexander quandles written by polynomials, called polynomial $($quandle$)$ cocycles, that can be used for such explicit calculations of the invariant. Such cocycles were first constructed in [Mochi03], and studied in [Ame06,Mochi05]. They have been extensively used in applications. The following formulas are found in [Ame06].

• $f(x,y)=(x-y)^{p^{n}}$ 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 $(t^{(p^{m_1}+p^{m_2}+p^{m_3})}-1)$.

More general formula for $n$-cocycles are given in [Ame06]. We cite (a slightly simplified version of) her result:

Proposition 4.4 ([Ame06])   Consider an Alexander quandle $X=\mathbb{Z}_p[t,t^{-1}]/h(t)=A$. Let $a_i=p^{m_i}$, for $i=1,\ldots,n-1$, where $p$ is a prime and $m_i$ are non-negative integers. For a positive integer $n$, let $f:X^n \rightarrow A$ be defined by

$$f(x_{1},x_{2},\ldots,x_{n}) =(x_{1}-x_{2})^{a_1}(x_{2}-x_{3})^{a_2}\cdots(x_{n-1}-x_{n})^{a_{n-1}}x_n^{a_n}.$$

Then $f$ is an $n$-cocycle $(\in Z^n_{\rm Q}(X;A))$, $(1)$ if $a_n=0$, or $(2)$ $a_n=p^{m_n}$ $($for a positive integer $m_n)$ and $g(t)$ divides $1-t^a$, where $a=a_1+a_2+\cdots+a_{n-1}+a_n$.

More specific polynomials and Alexander quandles considered are as follows. Non-triviality of quandle cocycle invariants for some of the cocycles below are obtained in [Ame06,StSm].

• $2$-cocycles:
  • $\mathbb{Z}_2$ coefficients:
    • $f(x,y)=(x-y)^{(2^m)} y$. Alexander quandle $\mathbb{Z}_2[t, t^{-1}]/(t^2+t+1)$ has a non-trivial $2$-cocycle $f(x,y)=(x-y)^2 y$. This $4$ element quandle is well-known, and for example, isomorphic to the quandle consisting of $120$ degree rotations of a regular tetrahedron. It is known to have $2$-dimensional cohomology group $\mathbb{Z}_2$ with $\mathbb{Z}_2$ coefficient [CJKLS03]. The invariant values for the coefficient $\mathbb{Z}_2[t, t^{-1}]/(t^2+t+1)$ are all of the form $k[4 + 12u^{(t+1)}]$, so we conjecture that it is always the case. It is also an interesting problem to characterize this invariant.
    • $f(x,y)= (x-y)^{(2^2)} y$. The quandle must be mod $g(t)$ where $g(t)$ divides $t^5-1$, but $t^5-1$ is factored into prime polynomials $(t+1)(t^4+t^3+t^2+t^2+1)$ mod $2$, so we set $g(t)=t^4+t^3+t^2+t^2+1$. The Alexander quandle we use in this case, thus, is $\mathbb{Z}_2[t, t^{-1}]/(t^4+t^3+t^2+t+1)$ with $2$-cocycle: $f(x,y)= (x-y)^{(2^2)} y$, which gives non-trivial invariants.
    • $f(x,y)= (x-y)^{(2^3)} y$. We factor $t^9-1$ mod $2$ to $(t+1)(t^2+t+1)(t^6+t^3+1)$, so we try Alexander quandle $\mathbb{Z}_2[t, t^{-1}]/(t^6+t^3+1)$, which gives non-trivial invariants.
  • $\mathbb{Z}_3$ coefficients:
    • $f(x,y)=(x-y)^3 y$. We have $t^4-1=(t+1)(t+2)(t^2+1)$ mod $3$, so we try the quandle $\mathbb{Z}_3[t, t^{-1}]/(t^2+1)$ which gives non-trivial invariants.
• $3$-cocycles:
  • $\mathbb{Z}_2$ coefficients: $f(x,y,z) = (x-y) (y-z)^2$. The cocycle is non-trivial for the quandle: $\mathbb{Z}_2[t, t^{-1}]/(t^2+t+1)$.
  • $\mathbb{Z}_3$ coefficients: $f(x,y,z) = (x-y) (y-z)^3$.

    The cocycle is non-trivial for the quandles: $\mathbb{Z}_3[t, t^{-1}]/(t^2+1)$, $\mathbb{Z}_3[t, t^{-1}]/(t^2-t+1)$.

  • $\mathbb{Z}_5$ coefficients: $f(x,y,z) = (x-y) (y-z)^5$. The cocycle is non-trivial for the quandle: $\mathbb{Z}_5[t, t^{-1}]/(t^2-t+1)$.
  • $\mathbb{Z}_7$ coefficients: $f(x,y,z) = (x-y) (y-z)^7$. The cocycle is non-trivial for the quandle: $\mathbb{Z}_7[t, t^{-1}]/(t^2-t+1)$.

There is another type of $3$-cocycles Mochizuki constructed specifically 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 $p$.