| Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,1,2,1,2,2,0,0,0,0,0,0,1,-1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.993'] |
| Arrow polynomial of the knot is: -4*K1**2 - 6*K1*K2 + 3*K1 + 2*K2 + 3*K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.586', '6.590', '6.958', '6.987', '6.991', '6.993', '6.999', '6.1054', '6.1065', '6.1096', '6.1168', '6.1182'] |
| Outer characteristic polynomial of the knot is: t^7+40t^5+40t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.993'] |
| 2-strand cable arrow polynomial of the knot is: -304*K1**4 + 32*K1**3*K2*K3 - 96*K1**3*K3 - 1200*K1**2*K2**2 - 192*K1**2*K2*K4 + 2136*K1**2*K2 - 96*K1**2*K3**2 - 2120*K1**2 + 96*K1*K2**3*K3 - 320*K1*K2**2*K3 - 96*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 2720*K1*K2*K3 + 600*K1*K3*K4 + 32*K1*K4*K5 - 112*K2**4 - 240*K2**2*K3**2 - 56*K2**2*K4**2 + 464*K2**2*K4 - 1758*K2**2 + 296*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 32*K3**2*K6 - 1148*K3**2 - 344*K4**2 - 84*K5**2 - 18*K6**2 + 1782 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.993'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11447', 'vk6.11742', 'vk6.12761', 'vk6.13104', 'vk6.20325', 'vk6.21666', 'vk6.27629', 'vk6.29173', 'vk6.31198', 'vk6.31537', 'vk6.32366', 'vk6.32781', 'vk6.39049', 'vk6.41309', 'vk6.45805', 'vk6.47480', 'vk6.52200', 'vk6.52461', 'vk6.53031', 'vk6.53351', 'vk6.57184', 'vk6.58395', 'vk6.61798', 'vk6.62919', 'vk6.63766', 'vk6.63876', 'vk6.64194', 'vk6.64380', 'vk6.66793', 'vk6.67661', 'vk6.69433', 'vk6.70155', 'vk6.81999', 'vk6.82730', 'vk6.84323', 'vk6.85647', 'vk6.86540', 'vk6.87557', 'vk6.88266', 'vk6.89409'] |
| 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 | O1O2O3O4U1U5O6U3O5U4U6U2 |
| R3 orbit | {'O1O2O3O4U1U5U2O6O5U4U3U6', 'O1O2O3O4U1U5O6U3O5U4U6U2'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U3U5U1O6U2O5U6U4 |
| Gauss code of K* | O1O2O3U4U3U5U1O4O6U2O5U6 |
| Gauss code of -K* | O1O2O3U4O5U2O4O6U3U5U1U6 |
| 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 -3 1 0 1 0 1],[ 3 0 3 1 2 2 2],[-1 -3 0 -1 0 -1 1],[ 0 -1 1 0 0 0 1],[-1 -2 0 0 0 -1 0],[ 0 -2 1 0 1 0 1],[-1 -2 -1 -1 0 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 0 -3],[-1 0 1 0 -1 -1 -3],[-1 -1 0 0 -1 -1 -2],[-1 0 0 0 0 -1 -2],[ 0 1 1 0 0 0 -1],[ 0 1 1 1 0 0 -2],[ 3 3 2 2 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,0,3,-1,0,1,1,3,0,1,1,2,0,1,2,0,1,2] |
| Phi over symmetry | [-3,0,0,1,1,1,1,2,1,2,2,0,0,0,0,0,0,1,-1,0,0] |
| Phi of -K | [-3,0,0,1,1,1,1,2,1,2,2,0,0,0,0,0,0,1,-1,0,0] |
| Phi of K* | [-1,-1,-1,0,0,3,-1,0,0,0,2,0,0,0,1,0,1,2,0,1,2] |
| Phi of -K* | [-3,0,0,1,1,1,1,2,2,2,3,0,0,1,1,1,1,1,0,0,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-3t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+16w^2z+21w |
| Inner characteristic polynomial | t^6+28t^4+20t^2 |
| Outer characteristic polynomial | t^7+40t^5+40t^3+3t |
| Flat arrow polynomial | -4*K1**2 - 6*K1*K2 + 3*K1 + 2*K2 + 3*K3 + 3 |
| 2-strand cable arrow polynomial | -304*K1**4 + 32*K1**3*K2*K3 - 96*K1**3*K3 - 1200*K1**2*K2**2 - 192*K1**2*K2*K4 + 2136*K1**2*K2 - 96*K1**2*K3**2 - 2120*K1**2 + 96*K1*K2**3*K3 - 320*K1*K2**2*K3 - 96*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 2720*K1*K2*K3 + 600*K1*K3*K4 + 32*K1*K4*K5 - 112*K2**4 - 240*K2**2*K3**2 - 56*K2**2*K4**2 + 464*K2**2*K4 - 1758*K2**2 + 296*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 32*K3**2*K6 - 1148*K3**2 - 344*K4**2 - 84*K5**2 - 18*K6**2 + 1782 |
| 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}], [{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}], [{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}, {4, 5}, {3}, {2}, {1}], [{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 |