| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,3,3,2,0,1,2,2,0,1,1,0,1,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.671'] |
| Arrow polynomial of the knot is: -6*K1**2 - 2*K1*K2 + K1 + 3*K2 + K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.323', '6.380', '6.444', '6.472', '6.523', '6.579', '6.592', '6.595', '6.609', '6.614', '6.620', '6.644', '6.648', '6.669', '6.671', '6.681', '6.693', '6.724', '6.725', '6.757', '6.766', '6.785', '6.786', '6.797', '6.798', '6.816', '6.833', '6.972', '6.978', '6.1056', '6.1064', '6.1066', '6.1087', '6.1094', '6.1273', '6.1277', '6.1282', '6.1295', '6.1300', '6.1313', '6.1344', '6.1353', '6.1354'] |
| Outer characteristic polynomial of the knot is: t^7+56t^5+76t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.671'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 160*K1**4*K2 - 592*K1**4 + 96*K1**3*K2*K3 - 576*K1**2*K2**2 + 976*K1**2*K2 - 112*K1**2*K3**2 - 372*K1**2 + 568*K1*K2*K3 + 96*K1*K3*K4 + 8*K1*K4*K5 - 88*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 64*K2**2*K4 - 374*K2**2 + 40*K2*K3*K5 + 8*K2*K4*K6 - 168*K3**2 - 42*K4**2 - 12*K5**2 - 2*K6**2 + 440 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.671'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13930', 'vk6.13950', 'vk6.14025', 'vk6.14044', 'vk6.14997', 'vk6.15017', 'vk6.15118', 'vk6.15137', 'vk6.16540', 'vk6.16633', 'vk6.17455', 'vk6.17477', 'vk6.17489', 'vk6.23470', 'vk6.23809', 'vk6.23964', 'vk6.23976', 'vk6.23997', 'vk6.24009', 'vk6.24120', 'vk6.25998', 'vk6.26382', 'vk6.33748', 'vk6.33828', 'vk6.35052', 'vk6.35615', 'vk6.36273', 'vk6.36371', 'vk6.37605', 'vk6.37692', 'vk6.38882', 'vk6.41082', 'vk6.43418', 'vk6.44587', 'vk6.45647', 'vk6.53882', 'vk6.54428', 'vk6.54786', 'vk6.54874', 'vk6.55602', 'vk6.56435', 'vk6.58230', 'vk6.59976', 'vk6.60095', 'vk6.60107', 'vk6.60180', 'vk6.62793', 'vk6.65032'] |
| The R3 orbit of minmal crossing diagrams contains: |
| The diagrammatic symmetry type of this knot is c. |
| The reverse -K is |
| The mirror image K* is |
| The reversed mirror image -K* is |
| The fillings (up to the first 10) associated to the algebraic genus: |
| Or click here to check the fillings |
| invariant | value |
|---|---|
| Gauss code | O1O2O3O4U2O5U6U3O6U4U1U5 |
| R3 orbit | {'O1O2O3U1O4O5U6U3O6U2U4U5', 'O1O2O3O4U2O5U4U6U3O6U1U5', 'O1O2O3O4U2O5U6U3O6U4U1U5'} |
| R3 orbit length | 3 |
| Gauss code of -K | O1O2O3O4U5U4U1O6U2U6O5U3 |
| Gauss code of K* | O1O2O3U2U4U5U1O4U3O6O5U6 |
| Gauss code of -K* | O1O2O3U4O5O4U1O6U3U5U6U2 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 2 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -1 -2 0 1 3 -1],[ 1 0 -2 1 2 3 0],[ 2 2 0 1 2 2 1],[ 0 -1 -1 0 0 1 0],[-1 -2 -2 0 0 1 -1],[-3 -3 -2 -1 -1 0 -3],[ 1 0 -1 0 1 3 0]] |
| Primitive based matrix | [[ 0 3 1 0 -1 -1 -2],[-3 0 -1 -1 -3 -3 -2],[-1 1 0 0 -1 -2 -2],[ 0 1 0 0 0 -1 -1],[ 1 3 1 0 0 0 -1],[ 1 3 2 1 0 0 -2],[ 2 2 2 1 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,1,1,2,1,1,3,3,2,0,1,2,2,0,1,1,0,1,2] |
| Phi over symmetry | [-3,-1,0,1,1,2,1,1,3,3,2,0,1,2,2,0,1,1,0,1,2] |
| Phi of -K | [-2,-1,-1,0,1,3,-1,0,1,1,3,0,0,0,1,1,1,1,1,2,1] |
| Phi of K* | [-3,-1,0,1,1,2,1,2,1,1,3,1,0,1,1,0,1,1,0,-1,0] |
| Phi of -K* | [-2,-1,-1,0,1,3,1,2,1,2,2,0,0,1,3,1,2,3,0,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 7z+15 |
| Enhanced Jones-Krushkal polynomial | 7w^2z+15w |
| Inner characteristic polynomial | t^6+40t^4+47t^2 |
| Outer characteristic polynomial | t^7+56t^5+76t^3 |
| Flat arrow polynomial | -6*K1**2 - 2*K1*K2 + K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -64*K1**4*K2**2 + 160*K1**4*K2 - 592*K1**4 + 96*K1**3*K2*K3 - 576*K1**2*K2**2 + 976*K1**2*K2 - 112*K1**2*K3**2 - 372*K1**2 + 568*K1*K2*K3 + 96*K1*K3*K4 + 8*K1*K4*K5 - 88*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 64*K2**2*K4 - 374*K2**2 + 40*K2*K3*K5 + 8*K2*K4*K6 - 168*K3**2 - 42*K4**2 - 12*K5**2 - 2*K6**2 + 440 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]] |
| If K is slice | False |