Twisted Quandle 2-Cocycle Knot Invariants

This page contains values for quandle 2-cocycle knot invariants for the twisted case. The 2-cocycles used were calculated with Maple by solving system of equations over Z_p[t,t^(-1)]/(h(t)), where p is prime. In this case, the coefficient group, A, is Z_p[t,t^(-1)]/(h(t)). For more information see section 2.7 in the background section. A typical value produced is [[6, 0], [2, x[12] t + (x[6] + x[7] + x[1]) t + x[7]],[2, x[11] t² + (x[6] + x[7] + x[1]) t + x[1] + x[5]], [2, (x[12] + x[11]) t² + x[7] + x[5] + x[1]]]. The value of the invariant is represented as a list of two element lists. The first element of the two element list is the number of colorings that produced the second term as its state sum contribution. The second term is an element of the coefficient group. The x[i] are free variables that take values from Z_p. These free variables are introduced from solving the twisted cocycle conditions and are carried over to the value the invariant takes.

We computed the invariants for knots in the table up to and including 8 crossings. The quandles were taken from the table of quandles that appeared in the appendix of [CKS00]. We used only quandles that had nontrivial torsion part in one of its homology groups. There was 1 three element quandle, 3 four element quandles, 6 five element quandles, and 25 six element quandles. The values of the invariant were calculated for every quandle of these orders with the coefficient groups (A=)Z_p[t,t^(-1)]/(h(t)) where p is in {2,3,5} and h(t) is in {t+1, t^2+1, t^2+t+1, t^3+t+1, t^3+t^2+1} and where p is in {3,5} where h(t)=t^2+2t+2.

If one of the quandles of a particular order for a given coefficient group produced non-trivial values in our calculations, then the section is marked "non-trivial."

3 element quandle

4 element quandles

5 element quandles

6 element quandles