Untwisted Quandle 3-Cocycle Knot Invariants

This page contains values for quandle 3-cocycle knot invariants for the untwisted case. The 3-cocycles used were calculated with Maple by solving an overdetermined system of equations over Z_p, where p is prime. In this case, the coefficient group, A, is Z_p. A typical value produced is either an integer value representing the number of colorings or a polynomial of the form, 9+18u^(t_26+t_21+t_22). For more information see section 2.5 in the background section. t_i (for integer i) is a free variable obtained from solving the cocycle conditions. These free variables are carried over to the value the invariant takes. Therefore, 9+18u^(t_26+t_21+t_22) is a family of values for different 3-cocycles.

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_2, Z_3, and Z_5.

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