| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,0,2,3,1,0,1,2,1,-1,-1,0,-1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1042'] |
| 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+68t^5+335t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1042'] |
| 2-strand cable arrow polynomial of the knot is: -400*K1**4 - 160*K1**3*K3 + 768*K1**2*K2**5 - 3840*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 3616*K1**2*K2**3 - 4304*K1**2*K2**2 - 64*K1**2*K2*K4 + 3568*K1**2*K2 - 48*K1**2*K3**2 - 2644*K1**2 + 2720*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 416*K1*K2**2*K3 + 2496*K1*K2*K3 + 80*K1*K3*K4 - 864*K2**6 + 384*K2**4*K4 - 1336*K2**4 - 544*K2**2*K3**2 - 24*K2**2*K4**2 + 424*K2**2*K4 - 256*K2**2 + 32*K2*K3*K5 - 636*K3**2 - 42*K4**2 + 1712 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1042'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4631', 'vk6.4896', 'vk6.6041', 'vk6.6560', 'vk6.8084', 'vk6.8459', 'vk6.9460', 'vk6.9839', 'vk6.20284', 'vk6.21615', 'vk6.27564', 'vk6.29128', 'vk6.38977', 'vk6.41224', 'vk6.45748', 'vk6.47443', 'vk6.48669', 'vk6.48852', 'vk6.49391', 'vk6.49638', 'vk6.50677', 'vk6.50852', 'vk6.51150', 'vk6.51373', 'vk6.57133', 'vk6.58325', 'vk6.61743', 'vk6.62884', 'vk6.66758', 'vk6.67642', 'vk6.69412', 'vk6.70136'] |
| 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 | O1O2O3O4U5U6O5U1O6U2U3U4 |
| R3 orbit | {'O1O2O3O4U5U6O5U1O6U2U3U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U2U3O5U4O6U5U6 |
| Gauss code of K* | O1O2O3U4U1U2U3O5O6U5O4U6 |
| Gauss code of -K* | O1O2O3U4O5U6O4O6U1U2U3U5 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 3 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -2 -1 1 3 -1 0],[ 2 0 0 1 2 2 3],[ 1 0 0 1 2 0 2],[-1 -1 -1 0 1 -2 0],[-3 -2 -2 -1 0 -4 -2],[ 1 -2 0 2 4 0 0],[ 0 -3 -2 0 2 0 0]] |
| Primitive based matrix | [[ 0 3 1 0 -1 -1 -2],[-3 0 -1 -2 -2 -4 -2],[-1 1 0 0 -1 -2 -1],[ 0 2 0 0 -2 0 -3],[ 1 2 1 2 0 0 0],[ 1 4 2 0 0 0 -2],[ 2 2 1 3 0 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,1,1,2,1,2,2,4,2,0,1,2,1,2,0,3,0,0,2] |
| Phi over symmetry | [-3,-1,0,1,1,2,1,1,0,2,3,1,0,1,2,1,-1,-1,0,-1,1] |
| Phi of -K | [-2,-1,-1,0,1,3,-1,1,-1,2,3,0,1,0,0,-1,1,2,1,1,1] |
| Phi of K* | [-3,-1,0,1,1,2,1,1,0,2,3,1,0,1,2,1,-1,-1,0,-1,1] |
| Phi of -K* | [-2,-1,-1,0,1,3,0,2,3,1,2,0,2,1,2,0,2,4,0,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 5z+11 |
| Enhanced Jones-Krushkal polynomial | -6w^4z^2+6w^3z^2-10w^3z+15w^2z+11w |
| Inner characteristic polynomial | t^6+52t^4+242t^2 |
| Outer characteristic polynomial | t^7+68t^5+335t^3+6t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -400*K1**4 - 160*K1**3*K3 + 768*K1**2*K2**5 - 3840*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 3616*K1**2*K2**3 - 4304*K1**2*K2**2 - 64*K1**2*K2*K4 + 3568*K1**2*K2 - 48*K1**2*K3**2 - 2644*K1**2 + 2720*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 416*K1*K2**2*K3 + 2496*K1*K2*K3 + 80*K1*K3*K4 - 864*K2**6 + 384*K2**4*K4 - 1336*K2**4 - 544*K2**2*K3**2 - 24*K2**2*K4**2 + 424*K2**2*K4 - 256*K2**2 + 32*K2*K3*K5 - 636*K3**2 - 42*K4**2 + 1712 |
| Genus of based matrix | 2 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{1, 6}, {5}, {4}, {3}, {2}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{5, 6}, {4}, {3}, {2}, {1}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {1, 5}, {4}, {3}, {2}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]] |
| If K is slice | False |