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.
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.
The integer Lq_X (K) of a knot K for a finite quandle X is the ceiling of
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)
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 )
8 additional larger quandles,
Q[i], i=16, 18, 19, 20, 22, 23, 25
of a family of
26 quandles that distinguish all 2977 knots.
None of the last 8 larger quandles is Alexander.
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.