| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,2,3,1,1,1,2,1,-1,-1,1,1,1,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.974'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['4.2', '6.303', '6.338', '6.381', '6.432', '6.468', '6.558', '6.583', '6.597', '6.607', '6.634', '6.637', '6.643', '6.654', '6.667', '6.701', '6.709', '6.712', '6.718', '6.728', '6.767', '6.801', '6.825', '6.827', '6.974', '6.994', '6.1042', '6.1061', '6.1069', '6.1181', '6.1271', '6.1286', '6.1287', '6.1289', '6.1297', '6.1337', '6.1355'] |
| Outer characteristic polynomial of the knot is: t^7+48t^5+116t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.974'] |
| 2-strand cable arrow polynomial of the knot is: -32*K1**4 + 96*K1**2*K2**3 - 576*K1**2*K2**2 + 688*K1**2*K2 - 636*K1**2 + 64*K1*K2**3*K3 + 648*K1*K2*K3 + 16*K1*K3*K4 + 8*K1*K4*K5 - 864*K2**6 + 576*K2**4*K4 - 472*K2**4 - 112*K2**2*K3**2 - 88*K2**2*K4**2 + 432*K2**2*K4 - 16*K2**2 + 56*K2*K3*K5 + 8*K2*K4*K6 - 208*K3**2 - 54*K4**2 - 12*K5**2 + 476 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.974'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70491', 'vk6.70493', 'vk6.70544', 'vk6.70548', 'vk6.70683', 'vk6.70691', 'vk6.70798', 'vk6.70802', 'vk6.70968', 'vk6.70972', 'vk6.71041', 'vk6.71048', 'vk6.71185', 'vk6.71189', 'vk6.71268', 'vk6.71270', 'vk6.73787', 'vk6.73790', 'vk6.73923', 'vk6.73927', 'vk6.74603', 'vk6.75727', 'vk6.75732', 'vk6.76092', 'vk6.78738', 'vk6.78743', 'vk6.79036', 'vk6.80347', 'vk6.81088', 'vk6.81092', 'vk6.81187', 'vk6.81221', 'vk6.81778', 'vk6.87299', 'vk6.87886', 'vk6.87890', 'vk6.87895', 'vk6.88420', 'vk6.88422', 'vk6.89010'] |
| 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 | O1O2O3O4U1U2O5U4O6U5U3U6 |
| R3 orbit | {'O1O2O3O4U1U2U3O5O6U4U5U6', 'O1O2O3O4U1U2O5U4O6U5U3U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U2U6O5U1O6U3U4 |
| Gauss code of K* | O1O2O3U4U5U2U6O4O5U1O6U3 |
| Gauss code of -K* | O1O2O3U1O4U3O5O6U4U2U5U6 |
| 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 -3 -1 1 1 0 2],[ 3 0 1 3 2 1 1],[ 1 -1 0 2 1 1 1],[-1 -3 -2 0 -1 1 2],[-1 -2 -1 1 0 1 1],[ 0 -1 -1 -1 -1 0 1],[-2 -1 -1 -2 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -1 -3],[-2 0 -1 -2 -1 -1 -1],[-1 1 0 1 1 -1 -2],[-1 2 -1 0 1 -2 -3],[ 0 1 -1 -1 0 -1 -1],[ 1 1 1 2 1 0 -1],[ 3 1 2 3 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,1,3,1,2,1,1,1,-1,-1,1,2,-1,2,3,1,1,1] |
| Phi over symmetry | [-3,-1,0,1,1,2,1,1,2,3,1,1,1,2,1,-1,-1,1,1,1,2] |
| Phi of -K | [-3,-1,0,1,1,2,1,2,1,2,4,0,0,1,2,2,2,1,1,-1,0] |
| Phi of K* | [-2,-1,-1,0,1,3,-1,0,1,2,4,-1,2,0,1,2,1,2,0,2,1] |
| Phi of -K* | [-3,-1,0,1,1,2,1,1,2,3,1,1,1,2,1,-1,-1,1,1,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 3z+7 |
| Enhanced Jones-Krushkal polynomial | 8w^4z-10w^3z+8w^3+5w^2z-w |
| Inner characteristic polynomial | t^6+32t^4+17t^2 |
| Outer characteristic polynomial | t^7+48t^5+116t^3 |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -32*K1**4 + 96*K1**2*K2**3 - 576*K1**2*K2**2 + 688*K1**2*K2 - 636*K1**2 + 64*K1*K2**3*K3 + 648*K1*K2*K3 + 16*K1*K3*K4 + 8*K1*K4*K5 - 864*K2**6 + 576*K2**4*K4 - 472*K2**4 - 112*K2**2*K3**2 - 88*K2**2*K4**2 + 432*K2**2*K4 - 16*K2**2 + 56*K2*K3*K5 + 8*K2*K4*K6 - 208*K3**2 - 54*K4**2 - 12*K5**2 + 476 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {4, 5}, {1, 2}], [{6}, {4, 5}, {3}, {1, 2}]] |
| If K is slice | False |