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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