Untwisted Quandle 2-Cocycle Knot Invariants

This page contains values for quandle 2-cocycle knot invariants for the untwisted case. The 2-cocycles used were calculated with Maple by solving an overdetermined system of equations over _p, where p is prime. In this case, the coefficient group, A, is _p. For more information see section 2.4 in the backgroud section. A typical value produced is either an integer value representing the number of colorings or a polynomial of the form, 4+12u^{(t₁₀+t₁₅+t₁₄)}. t_{i p} is a free variable obtained from solving the cocycle conditions. These free variables are carried over to the value the invariant takes. Therefore, 4+12u^{(t₁₀+t₁₅+t₁₄)} is a family of values for different 2-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=) ₂, ₃, and 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