| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,3,1,1,3,2,0,1,1,1,0,2,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1286'] |
| 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+58t^5+111t^3+13t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1286'] |
| 2-strand cable arrow polynomial of the knot is: -448*K1**4*K2**2 + 800*K1**4*K2 - 800*K1**4 + 192*K1**3*K2*K3 - 96*K1**3*K3 - 832*K1**2*K2**4 + 2560*K1**2*K2**3 - 9520*K1**2*K2**2 - 480*K1**2*K2*K4 + 7248*K1**2*K2 - 3792*K1**2 + 960*K1*K2**3*K3 - 448*K1*K2**2*K3 - 128*K1*K2**2*K5 + 5728*K1*K2*K3 + 8*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 1592*K2**4 - 272*K2**2*K3**2 - 8*K2**2*K4**2 + 688*K2**2*K4 - 1528*K2**2 + 64*K2*K3*K5 - 744*K3**2 - 18*K4**2 + 2456 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1286'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4721', 'vk6.5038', 'vk6.6248', 'vk6.6698', 'vk6.8218', 'vk6.8656', 'vk6.9600', 'vk6.9927', 'vk6.20293', 'vk6.21626', 'vk6.27589', 'vk6.29141', 'vk6.39007', 'vk6.41255', 'vk6.45775', 'vk6.47452', 'vk6.48761', 'vk6.48964', 'vk6.49564', 'vk6.49776', 'vk6.50771', 'vk6.50977', 'vk6.51252', 'vk6.51457', 'vk6.57152', 'vk6.58336', 'vk6.61778', 'vk6.62897', 'vk6.66773', 'vk6.67649', 'vk6.69421', 'vk6.70143'] |
| 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 | O1O2O3U1O4O5U6U4O6U2U3U5 |
| R3 orbit | {'O1O2O3U1O4O5U6U4O6U2U3U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U1U2O5U6U5O4O6U3 |
| Gauss code of K* | O1O2O3U4U1U2O4U5U3O6O5U6 |
| Gauss code of -K* | O1O2O3U4O5O4U1U5O6U2U3U6 |
| 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 0 3 -1],[ 2 0 1 2 0 2 2],[ 1 -1 0 1 1 3 0],[-1 -2 -1 0 1 2 -2],[ 0 0 -1 -1 0 0 0],[-3 -2 -3 -2 0 0 -3],[ 1 -2 0 2 0 3 0]] |
| Primitive based matrix | [[ 0 3 1 0 -1 -1 -2],[-3 0 -2 0 -3 -3 -2],[-1 2 0 1 -1 -2 -2],[ 0 0 -1 0 -1 0 0],[ 1 3 1 1 0 0 -1],[ 1 3 2 0 0 0 -2],[ 2 2 2 0 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,1,1,2,2,0,3,3,2,-1,1,2,2,1,0,0,0,1,2] |
| Phi over symmetry | [-3,-1,0,1,1,2,0,3,1,1,3,2,0,1,1,1,0,2,0,-1,0] |
| Phi of -K | [-2,-1,-1,0,1,3,-1,0,2,1,3,0,1,0,1,0,1,1,2,3,0] |
| Phi of K* | [-3,-1,0,1,1,2,0,3,1,1,3,2,0,1,1,1,0,2,0,-1,0] |
| Phi of -K* | [-2,-1,-1,0,1,3,1,2,0,2,2,0,1,1,3,0,2,3,-1,0,2] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 6z^2+23z+23 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2-4w^3z+27w^2z+23w |
| Inner characteristic polynomial | t^6+42t^4+66t^2+1 |
| Outer characteristic polynomial | t^7+58t^5+111t^3+13t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -448*K1**4*K2**2 + 800*K1**4*K2 - 800*K1**4 + 192*K1**3*K2*K3 - 96*K1**3*K3 - 832*K1**2*K2**4 + 2560*K1**2*K2**3 - 9520*K1**2*K2**2 - 480*K1**2*K2*K4 + 7248*K1**2*K2 - 3792*K1**2 + 960*K1*K2**3*K3 - 448*K1*K2**2*K3 - 128*K1*K2**2*K5 + 5728*K1*K2*K3 + 8*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 1592*K2**4 - 272*K2**2*K3**2 - 8*K2**2*K4**2 + 688*K2**2*K4 - 1528*K2**2 + 64*K2*K3*K5 - 744*K3**2 - 18*K4**2 + 2456 |
| 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}], [{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}, {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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]] |
| If K is slice | False |