**Home**
**Quandles**
**Programs and Source Code**
**Microscope Images / Video**
**Ray tracing**
**Links**
**Hacks**
**Resume**

**..**

# Small Connected Quandles and Their Knot Colorings: Project Page

### Small Connected Quandles and Their Knot Colorings: Project Page

This website contains computational results on small connected quandles and their knot colorings.
These results are obtained during 2011 in a research group consisting of
W. Edwin Clark,
Mohammed Elhamdadi,
Masahico Saito, and
Timothy Yeatman.

Saito's project page is located here.

## Latest results:

March, 2012 -- We can now distinguish all knots with 12 or fewer
crossings by quandle coloring using only 20 quandles! Data tables are
available!
September 16, 2011 -- We can now distinguish all knots with 12 or fewer
crossings by quandle coloring.
## Data Sources:

Cha, J. C.; Livingston, C., KnotInfo: Table of Knot Invariants,
http://www.indiana.edu/~knotinfo, May 26, 2011.
Vendramin, Leandro,
RIG - a GAP package for racks and quandles, May 22, 2011,
http://code.google.com/p/rig/.
We call the quandles in Vendramin's website ``rig quandles''.
Stoimenow, Alexander, Braid descriptions, June 5, 2011,
http://stoimenov.net/stoimeno/homepage/ptab/index.html

## Data Tables:

The following are coloring matrices obtained by W. Edwin Clark and Timothy Yeatman.
The (i,j)-entry of the matrix is the number of non-trivial colorings by quandle
Q[i] (given below) of the jth knot in the table.

2977 knots (1 through 12 crossings) colored by 20 connected quandles (GAP format).

2977 knots (1 through 12 crossings) colored by 20 connected quandles (MAPLE format).

20 connected quandles that distinguish knots with 12 or fewer crossings (GAP format).

20 connected quandles that distinguish knots with 12 or fewer crossings (MAPLE format).

verify knot colorings of 20 quandles with a C program (note:uses UN*X style '/' for paths).

Q[1]: order = 12, Inn(Q[1] = SmallGroup(60,5) = A5

Q[2]: order = 13, Inn(Q[2] = SmallGroup(52,3) = C13 : C4

Q[3]: order = 17, Inn(Q[3] = SmallGroup(136,12) = C17 : C8

Q[4]: order = 19, Inn(Q[4] = SmallGroup(57,1) = C19 : C3

Q[5]: order = 20, Inn(Q[5] = SmallGroup(120,34) = S5

Q[6]: order = 23, Inn(Q[6] = SmallGroup(253,1) = C23 : C11

Q[7]: order = 23, Inn(Q[7] = SmallGroup(506,1) = (C23 : C11) : C2

Q[8]: order = 24, Inn(Q[8] = SmallGroup(168,42) = PSL(3,2)

Q[9]: order = 25, Inn(Q[9] = SmallGroup(75,2) = (C5 x C5) : C3

Q[10]: order = 30, Inn(Q[10] = SmallGroup(120,34) = S5

Q[11]: order = 31, Inn(Q[11] = SmallGroup(155,1) = C31 : C5

Q[12]: order = 35, Inn(Q[12] = SmallGroup(70,3) = D70

Q[13]: order = 40, Inn(Q[13] = SmallGroup(320,1635) = ((C2 x C2 x C2 x C2) : C5) : C4

Q[14]: order = 42, Inn(Q[14] = SmallGroup(168,42) = PSL(3,2)

Q[15]: order = 48, Inn(Q[15] = SmallGroup(288,1025) = (A4 x A4) : C2

Q[16]: order = 60, Inn(Q[16] = SmallGroup(660,13) = PSL(2,11)

Q[17]: order = 72, Inn(Q[17] = SmallGroup(576,8652) = (A4 x A4) : C4

Q[18]: order = 72, Inn(Q[18] = SmallGroup(504,156) = PSL(2,8)

Q[19]: order = 84, Inn(Q[19] = SmallGroup(1512,779) = PSL(2,8) : C3

Q[20]: order = 90, Inn(Q[20] = SmallGroup(720,765) = A6 . C2

2977 knots (1 through 12 crossings) colored by 431 connected quandles from RIG
12965 knots (1 through 13 crossings) colored by 12 connected quandles (more to come...)
2977 knots (1 through 12 crossings) colored by 100 connected Galkin quandles of orders: 36,39,42,45,48,51,54,57,60,63,66,69,72,75,78
For easier viewing, see this version: 2977 knots (1 through 12 crossings) colored by 60 connected galkin quandles of orders 36 through 60