Quandle Colorings and Invariants of Knots in the Table

This website contains information on relations between quandle colorings and a few invariants of knots in the table.
This is a summary of observations made by W. Edwin Clark and Masahico Saito in 2012-2013.

Background

Backgound material on quandles is available at the website
Quandle Cocycle Knot Invariants.

Our computations use results on quandle knot colorings posted at the website
Small Connected Quandles and Their Knot Colorings.

Computational outputs

The integer Lq_X (K) of a knot K for a finite quandle X is the ceiling of log_b (Col_{X}(K)),
where Col_X (K) denotes the number of colorings of K by X, and b=|X| denotes the order of X.
The following files contain the maximum values MLq(K) of Lq_X (K)
for 2977 knots K in the table with 12 crossings or less, over a set of 439 quandles X.
These 439 quandles consist of 431 RIG quandles (see the website on small connected quandles and their knot colorings )
and 8 additional larger quandles, Q[i], i=16, 18, 19, 20, 22, 23, 25 from of a family of 26 quandles that distinguish all 2977 knots.
None of the last 8 larger quandles is Alexander.

Values of MLq(K) for all knots with 12 crossings or less
Knots with MLq(K) = 3
Knots with MLq(K) = 4
The first quandle X that give Lq_X(K) = 3
The first quandle X that give Lq_X(K) = 4
The first simple quandle of prime power order X that gives Lq_X(K) = 3
The first simple quandle of prime power order X that gives Lq_X(K) = 3 omitting knots with no such quandle
The matrix LM439[i,j] = Lq_X(i) (K(j)) where X(i) is the i-th quandle and K(j) is the j-th knot in the table
Knots with MLq=3 by simple quandles of prime power order and the unknotting number
Comparing MLq=3 by simple quandles of prime power order, the unknotting number, and the bridge index

Acknowledgement

We thank Chuck Livingston and Chad Musick for valuable conversations.
MS was supported in part by the National Science Foundation under Grant No. DMS-0900671.
Any opinions, findings, and conclusions or recommendations
expressed in this material are those of the author(s)
and do not necessarily reflect the views of the National Science Foundation.