| Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,1,1,3,3,1,1,2,2,1,1,1,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.667'] |
| 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+67t^5+32t^3+2t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.667'] |
| 2-strand cable arrow polynomial of the knot is: -1536*K1**4*K2**2 + 3008*K1**4*K2 - 3648*K1**4 + 832*K1**3*K2*K3 - 224*K1**3*K3 - 448*K1**2*K2**4 + 2144*K1**2*K2**3 - 8224*K1**2*K2**2 - 288*K1**2*K2*K4 + 6680*K1**2*K2 - 1516*K1**2 + 448*K1*K2**3*K3 - 704*K1*K2**2*K3 + 3944*K1*K2*K3 + 8*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 904*K2**4 - 144*K2**2*K3**2 - 8*K2**2*K4**2 + 424*K2**2*K4 - 1296*K2**2 + 16*K2*K3*K5 - 348*K3**2 - 18*K4**2 + 1800 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.667'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16905', 'vk6.17147', 'vk6.20518', 'vk6.21904', 'vk6.23297', 'vk6.23596', 'vk6.27963', 'vk6.29438', 'vk6.35311', 'vk6.35747', 'vk6.39367', 'vk6.41547', 'vk6.42816', 'vk6.43098', 'vk6.45936', 'vk6.47621', 'vk6.55060', 'vk6.55305', 'vk6.57387', 'vk6.58555', 'vk6.59456', 'vk6.59745', 'vk6.62046', 'vk6.63040', 'vk6.64901', 'vk6.65114', 'vk6.66932', 'vk6.67785', 'vk6.68210', 'vk6.68354', 'vk6.69539', 'vk6.70242'] |
| 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 | O1O2O3O4U2O5U6U1O6U3U4U5 |
| R3 orbit | {'O1O2O3O4U2O5U6U1O6U3U4U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U1U2O6U4U6O5U3 |
| Gauss code of K* | O1O2O3U4U5U1U2O5U3O6O4U6 |
| Gauss code of -K* | O1O2O3U4O5O4U1O6U2U3U6U5 |
| 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 -2 0 2 3 -1],[ 2 0 0 1 2 2 2],[ 2 0 0 1 2 2 2],[ 0 -1 -1 0 1 2 0],[-2 -2 -2 -1 0 1 -2],[-3 -2 -2 -2 -1 0 -3],[ 1 -2 -2 0 2 3 0]] |
| Primitive based matrix | [[ 0 3 2 0 -1 -2 -2],[-3 0 -1 -2 -3 -2 -2],[-2 1 0 -1 -2 -2 -2],[ 0 2 1 0 0 -1 -1],[ 1 3 2 0 0 -2 -2],[ 2 2 2 1 2 0 0],[ 2 2 2 1 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,0,1,2,2,1,2,3,2,2,1,2,2,2,0,1,1,2,2,0] |
| Phi over symmetry | [-3,-2,0,1,2,2,0,1,1,3,3,1,1,2,2,1,1,1,-1,-1,0] |
| Phi of -K | [-2,-2,-1,0,2,3,0,-1,1,2,3,-1,1,2,3,1,1,1,1,1,0] |
| Phi of K* | [-3,-2,0,1,2,2,0,1,1,3,3,1,1,2,2,1,1,1,-1,-1,0] |
| Phi of -K* | [-2,-2,-1,0,2,3,0,2,1,2,2,2,1,2,2,0,2,3,1,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 8z^2+29z+27 |
| Enhanced Jones-Krushkal polynomial | 8w^3z^2+29w^2z+27w |
| Inner characteristic polynomial | t^6+45t^4+5t^2 |
| Outer characteristic polynomial | t^7+67t^5+32t^3+2t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -1536*K1**4*K2**2 + 3008*K1**4*K2 - 3648*K1**4 + 832*K1**3*K2*K3 - 224*K1**3*K3 - 448*K1**2*K2**4 + 2144*K1**2*K2**3 - 8224*K1**2*K2**2 - 288*K1**2*K2*K4 + 6680*K1**2*K2 - 1516*K1**2 + 448*K1*K2**3*K3 - 704*K1*K2**2*K3 + 3944*K1*K2*K3 + 8*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 904*K2**4 - 144*K2**2*K3**2 - 8*K2**2*K4**2 + 424*K2**2*K4 - 1296*K2**2 + 16*K2*K3*K5 - 348*K3**2 - 18*K4**2 + 1800 |
| 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}], [{3, 6}, {5}, {4}, {2}, {1}], [{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}], [{4, 6}, {5}, {3}, {2}, {1}], [{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}, {3, 5}, {4}, {2}, {1}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}]] |
| If K is slice | False |