Database of Quandle Cocycle Knot Invariants

This page is the front end to our database. By checking the desired boxes you can create a table of values of different quandle cocycle knot invariants.

See the NOTES page for comments about the contents and use of the quandle cocycle knot invariant database.

Crossing Numbers of Knots

Select the crossing numbers of the prime knots that you want in your table.

8 and fewer 9 10 11 12


Knot Information

Check the boxes of the knot information that you want in your table.

Numbered Knot Name Braid Word Crossing Number Braid Length Number of Strands Alexander Polynomial


Quandle Cocycle Invariants with Alexander Quandles and Mochizuki Cocycles


Dihedral Quandles with Mochizuki 3-Cocycles f(x,y,z)=(x-y)[(2zp-yp)-(2z-y)p]/p mod p
R2 R3 R5 R7 R11
R13 R17 R19 R23 R29
R31 R37 R41 R43 R47

Cocycle values
Values of Mochizuki 3-cocycle formula f(x,y,z)=(x-y)[(2zp-yp)-(2z-y)p]/p mod p

Alexander Quandles with Mochizuki 2-cocycles f(x,y)=(x-y)p
2[t,t-1] / (t^3+1) 2[t,t-1] / (t^3+t+1) 2[t,t-1] / (t^3+t^2+1) 2[t,t-1] / (t^3+t^2+t+1)
3[t,t-1] / (t^2+1) 3[t,t-1] / (t^2+2) 3[t,t-1] / (t^2+t+1) 3[t,t-1] / (t^2+t+2) 3[t,t-1] / (t^2+2t+1)
3[t,t-1] / (t^2+2t+2)

Cocycle values
Values of Mochizuki 2-cocycle formula f(x,y)=(x-y)p
Alexander Quandles with Mochizuki 2-cocycles f(x,y)=(x-y)p * y
2[t,t-1] / (t^2-t+1) 3[t,t-1] / (t^2+1)

Cocycle values
Values of Mochizuki 2-cocycle formula f(x,y)=(x-y)p * y
Alexander Quandles with Mochizuki 2-cocycles f(x,y)=(x-y)p * y3
3[t,t-1] / (t^2+t+1) 3[t,t-1] / (t^2+2t+1)

Cocycle values
Values of Mochizuki 2-cocycle formula f(x,y)=(x-y)p * y3
Alexander Quandles with Mochizuki 2-cocycles f(x,y)=(x-y)p2 * y
2[t,t-1] / (t^4+t^3+t^2+t+1)

Values of Mochizuki 2-cocycle formula f(x,y)=(x-y)p2 * y
Alexander Quandles with Mochizuki 2-cocycles f(x,y)=(x-y)p3 * y
2[t,t-1] / (t^6+t^3+1)

Cocycle values
Values of Mochizuki 2-cocycle formula f(x,y)=(x-y)p3 * y
Alexander Quandles with Mochizuki 2-cocycles f(x,y)=(x-y)p2 * y2
2[t,t-1] / ((t^2+t+1)2 )

Values of Mochizuki 2-cocycle formula f(x,y)=(x-y)p2 * y2
Alexander Quandles with Mochizuki 2-cocycles f(x,y)=(x-y)p3 * y2
2[t,t-1] / (t^4+t^3+t^2+t+1)

Cocycle values
Values of Mochizuki 2-cocycle formula f(x,y)=(x-y)p3 * y2
Alexander Quandles with Mochizuki 2-cocycles f(x,y)=(x-y)p2 * y4
2[t,t-1] / (t^4+1)

Cocycle values
Values of Mochizuki 2-cocycle formula f(x,y)=(x-y)p2 * y4
Alexander Quandles with Mochizuki 2-cocycles f(x,y)=(x-y)p3 * y4
2[t,t-1] / (t^4+t^2+1)

Cocycle values
Values of Mochizuki 2-cocycle formula f(x,y)=(x-y)p3 * y4
Alexander Quandles with Mochizuki 3-cocycles f(x,y)=(x-y)*(y-z)p
2[t,t-1] / (t^2+t+1) 2[t,t-1] / (t^3+1) 2[t,t-1] / (t^3+t+1) 2[t,t-1] / (t^3+t^2+1) 2[t,t-1] / (t^3+t^2+t+1)
3[t,t-1] / (t^2+1) 3[t,t-1] / (t^2+2) 3[t,t-1] / (t^2+t+1) 3[t,t-1] / (t^2+t+2) 3[t,t-1] / (t^2+2t+1)
5[t,t-1] / (t^2-t+1) 7[t,t-1] / (t^2-t+1)

Cocycle values
Values of Mochizuki 3-cocycle formula f(x,y)=(x-y)*(y-z)p

Quandle Cocycle Invariants Quandles of 3-6 Elements and All Cocycles


Check the boxes of the quandles that you want in your table.

Quandles: 3 elements
Q3_1 Q3_2 Q3_3
Quandles: 4 elements
Q4_1 Q4_2 Q4_3 Q4_4 Q4_5 Q4_6 Q4_7
Quandles:5 elements
Q5_1 Q5_2 Q5_3 Q5_4 Q5_5 Q5_6 Q5_7
Q5_8 Q5_9 Q5_10 Q5_11 Q5_12 Q5_13 Q5_14
Q5_15 Q5_16 Q5_17 Q5_18 Q5_19 Q5_20 Q5_21
Q5_22
Quandles:6 elements
Q6_1 Q6_2 Q6_3 Q6_4 Q6_5 Q6_6 Q6_7
Q6_8 Q6_9 Q6_10 Q6_11 Q6_12 Q6_13 Q6_14
Q6_15 Q6_16 Q6_17 Q6_18 Q6_19 Q6_20 Q6_21
Q6_22 Q6_23 Q6_24 Q6_25 Q6_26 Q6_27 Q6_28
Q6_29 Q6_30 Q6_31 Q6_32 Q6_33 Q6_34 Q6_35
Q6_36 Q6_37 Q6_38 Q6_39 Q6_40 Q6_41 Q6_42
Q6_43 Q6_44 Q6_45 Q6_46 Q6_47 Q6_48 Q6_49
Q6_50 Q6_51 Q6_52 Q6_53 Q6_54 Q6_55 Q6_56
Q6_57 Q6_58 Q6_59 Q6_60 Q6_61 Q6_62 Q6_63
Q6_64 Q6_65 Q6_66 Q6_67 Q6_68 Q6_69 Q6_70
Q6_71 Q6_72 Q6_73

Cocycles

Check the boxes of the cocycles that you want in your table.

Untwisted case

Twisted case

All untwisted 2-cocycles All untwisted 3-cocycles All twisted 2-cocycles

Coefficient Group
2 3 5 7

Cocycle values
Values of the chosen cocycles

Quandle Information

Check the boxes of the quandle information that you want in your table.

Order Name Cayley Table Homology Groups Nontrivial torsion Type Alias