| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,1,1,2,4,1,1,1,2,-1,0,0,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1181'] |
| 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+52t^5+73t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1181'] |
| 2-strand cable arrow polynomial of the knot is: -96*K1**4 + 96*K1**2*K2**3 - 768*K1**2*K2**2 + 1072*K1**2*K2 - 1092*K1**2 + 96*K1*K2**3*K3 + 1216*K1*K2*K3 + 40*K1*K3*K4 + 8*K1*K4*K5 - 864*K2**6 + 576*K2**4*K4 - 888*K2**4 - 240*K2**2*K3**2 - 88*K2**2*K4**2 + 672*K2**2*K4 - 224*K2**2 + 192*K2*K3*K5 + 8*K2*K4*K6 - 472*K3**2 - 86*K4**2 - 52*K5**2 + 892 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1181'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72620', 'vk6.72625', 'vk6.72766', 'vk6.72774', 'vk6.73083', 'vk6.73090', 'vk6.73166', 'vk6.73169', 'vk6.73780', 'vk6.73783', 'vk6.73918', 'vk6.73919', 'vk6.75714', 'vk6.75719', 'vk6.75922', 'vk6.77869', 'vk6.77913', 'vk6.77925', 'vk6.78013', 'vk6.78722', 'vk6.78730', 'vk6.78927', 'vk6.80339', 'vk6.80343', 'vk6.81181', 'vk6.81184', 'vk6.81775', 'vk6.82481', 'vk6.87295', 'vk6.87900', 'vk6.88415', 'vk6.88417'] |
| 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 | O1O2O3O4U1U5U4O6O5U2U3U6 |
| R3 orbit | {'O1O2O3O4U1U5U4O6O5U2U3U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U2U3O6O5U1U6U4 |
| Gauss code of K* | O1O2O3U4U2O5O6O4U1U5U6U3 |
| Gauss code of -K* | O1O2O3U4U1O5O4O6U5U2U3U6 |
| 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 2 0 1],[ 3 0 2 3 1 2 2],[ 1 -2 0 1 1 0 1],[-1 -3 -1 0 1 -2 0],[-2 -1 -1 -1 0 -2 -1],[ 0 -2 0 2 2 0 1],[-1 -2 -1 0 1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -1 -3],[-2 0 -1 -1 -2 -1 -1],[-1 1 0 0 -1 -1 -2],[-1 1 0 0 -2 -1 -3],[ 0 2 1 2 0 0 -2],[ 1 1 1 1 0 0 -2],[ 3 1 2 3 2 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,1,3,1,1,2,1,1,0,1,1,2,2,1,3,0,2,2] |
| Phi over symmetry | [-3,-1,0,1,1,2,0,1,1,2,4,1,1,1,2,-1,0,0,0,0,0] |
| Phi of -K | [-3,-1,0,1,1,2,0,1,1,2,4,1,1,1,2,-1,0,0,0,0,0] |
| Phi of K* | [-2,-1,-1,0,1,3,0,0,0,2,4,0,-1,1,1,0,1,2,1,1,0] |
| Phi of -K* | [-3,-1,0,1,1,2,2,2,2,3,1,0,1,1,1,1,2,2,0,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 5z+11 |
| Enhanced Jones-Krushkal polynomial | 8w^4z-12w^3z+8w^3+9w^2z+3w |
| Inner characteristic polynomial | t^6+36t^4+20t^2 |
| Outer characteristic polynomial | t^7+52t^5+73t^3 |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -96*K1**4 + 96*K1**2*K2**3 - 768*K1**2*K2**2 + 1072*K1**2*K2 - 1092*K1**2 + 96*K1*K2**3*K3 + 1216*K1*K2*K3 + 40*K1*K3*K4 + 8*K1*K4*K5 - 864*K2**6 + 576*K2**4*K4 - 888*K2**4 - 240*K2**2*K3**2 - 88*K2**2*K4**2 + 672*K2**2*K4 - 224*K2**2 + 192*K2*K3*K5 + 8*K2*K4*K6 - 472*K3**2 - 86*K4**2 - 52*K5**2 + 892 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {2, 5}, {1, 4}]] |
| If K is slice | False |