| Min(phi) over symmetries of the knot is: [-4,-1,0,0,2,3,1,2,4,3,3,1,2,2,2,0,1,2,2,3,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.264'] |
| Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 3*K2 + K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.143', '6.158', '6.264', '6.282', '6.501'] |
| Outer characteristic polynomial of the knot is: t^7+101t^5+67t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.264'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 64*K1**4*K2 - 1024*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 256*K1**3*K2*K3 - 416*K1**3*K3 - 1664*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 2752*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 6176*K1**2*K2**2 + 160*K1**2*K2*K3**2 - 384*K1**2*K2*K4 + 6288*K1**2*K2 - 448*K1**2*K3**2 - 4568*K1**2 - 128*K1*K2**4*K3 + 2976*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1088*K1*K2**2*K3 - 160*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 6072*K1*K2*K3 + 752*K1*K3*K4 + 24*K1*K4*K5 + 16*K1*K5*K6 - 576*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 704*K2**4*K4 - 2608*K2**4 + 224*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 1568*K2**2*K3**2 - 280*K2**2*K4**2 + 1448*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 1802*K2**2 + 648*K2*K3*K5 + 56*K2*K4*K6 + 8*K2*K5*K7 + 24*K3**2*K6 - 1824*K3**2 - 406*K4**2 - 112*K5**2 - 30*K6**2 + 3532 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.264'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73703', 'vk6.73820', 'vk6.74195', 'vk6.74806', 'vk6.75624', 'vk6.75812', 'vk6.76358', 'vk6.76871', 'vk6.78603', 'vk6.78801', 'vk6.79224', 'vk6.79696', 'vk6.80245', 'vk6.80385', 'vk6.80699', 'vk6.81067', 'vk6.81610', 'vk6.81796', 'vk6.81920', 'vk6.82169', 'vk6.82297', 'vk6.82645', 'vk6.83187', 'vk6.84050', 'vk6.84210', 'vk6.84684', 'vk6.85005', 'vk6.86019', 'vk6.87745', 'vk6.88198', 'vk6.89398', 'vk6.89604'] |
| 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 | O1O2O3O4O5U1U3O6U4U5U2U6 |
| R3 orbit | {'O1O2O3O4O5U1U3O6U4U5U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U6U4U1U2O6U3U5 |
| Gauss code of K* | O1O2O3O4U5U3U6U1U2O5O6U4 |
| Gauss code of -K* | O1O2O3O4U1O5O6U3U4U5U2U6 |
| 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 -4 0 -1 0 2 3],[ 4 0 4 1 2 3 3],[ 0 -4 0 -2 0 2 3],[ 1 -1 2 0 1 2 2],[ 0 -2 0 -1 0 1 2],[-2 -3 -2 -2 -1 0 1],[-3 -3 -3 -2 -2 -1 0]] |
| Primitive based matrix | [[ 0 3 2 0 0 -1 -4],[-3 0 -1 -2 -3 -2 -3],[-2 1 0 -1 -2 -2 -3],[ 0 2 1 0 0 -1 -2],[ 0 3 2 0 0 -2 -4],[ 1 2 2 1 2 0 -1],[ 4 3 3 2 4 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,0,0,1,4,1,2,3,2,3,1,2,2,3,0,1,2,2,4,1] |
| Phi over symmetry | [-4,-1,0,0,2,3,1,2,4,3,3,1,2,2,2,0,1,2,2,3,1] |
| Phi of -K | [-4,-1,0,0,2,3,2,0,2,3,4,-1,0,1,2,0,0,0,1,1,0] |
| Phi of K* | [-3,-2,0,0,1,4,0,0,1,2,4,0,1,1,3,0,-1,0,0,2,2] |
| Phi of -K* | [-4,-1,0,0,2,3,1,2,4,3,3,1,2,2,2,0,1,2,2,3,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-t^3-t^2+t |
| Normalized Jones-Krushkal polynomial | 2z^2+15z+23 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+4w^3z^2-6w^3z+21w^2z+23w |
| Inner characteristic polynomial | t^6+71t^4+12t^2 |
| Outer characteristic polynomial | t^7+101t^5+67t^3+7t |
| Flat arrow polynomial | 8*K1**3 + 4*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -64*K1**4*K2**2 + 64*K1**4*K2 - 1024*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 256*K1**3*K2*K3 - 416*K1**3*K3 - 1664*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 2752*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 6176*K1**2*K2**2 + 160*K1**2*K2*K3**2 - 384*K1**2*K2*K4 + 6288*K1**2*K2 - 448*K1**2*K3**2 - 4568*K1**2 - 128*K1*K2**4*K3 + 2976*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1088*K1*K2**2*K3 - 160*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 6072*K1*K2*K3 + 752*K1*K3*K4 + 24*K1*K4*K5 + 16*K1*K5*K6 - 576*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 704*K2**4*K4 - 2608*K2**4 + 224*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 1568*K2**2*K3**2 - 280*K2**2*K4**2 + 1448*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 1802*K2**2 + 648*K2*K3*K5 + 56*K2*K4*K6 + 8*K2*K5*K7 + 24*K3**2*K6 - 1824*K3**2 - 406*K4**2 - 112*K5**2 - 30*K6**2 + 3532 |
| 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}, {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 |