| Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,0,1,2,1,3,0,0,1,0,0,1,1,1,2,-2] |
| Flat knots (up to 7 crossings) with same phi are :['6.1147'] |
| Arrow polynomial of the knot is: -2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + 2*K4 + 1 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.126', '6.195', '6.367', '6.438', '6.869', '6.872', '6.896', '6.1147'] |
| Outer characteristic polynomial of the knot is: t^7+45t^5+86t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1147'] |
| 2-strand cable arrow polynomial of the knot is: -576*K1**3*K3 - 192*K1**2*K2**2 + 288*K1**2*K2*K3**2 + 2976*K1**2*K2 - 2240*K1**2*K3**2 - 96*K1**2*K3*K5 - 48*K1**2*K6**2 - 4932*K1**2 + 64*K1*K2**3*K3 - 1120*K1*K2**2*K3 + 288*K1*K2*K3**3 - 288*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 6200*K1*K2*K3 - 32*K1*K2*K5*K6 - 64*K1*K3**2*K5 + 2832*K1*K3*K4 + 192*K1*K4*K5 + 200*K1*K5*K6 + 80*K1*K6*K7 - 64*K2**4 - 32*K2**3*K6 - 880*K2**2*K3**2 - 8*K2**2*K4**2 + 568*K2**2*K4 - 16*K2**2*K6**2 - 3614*K2**2 - 128*K2*K3**2*K4 + 1032*K2*K3*K5 + 224*K2*K4*K6 + 16*K2*K5*K7 + 32*K2*K6*K8 - 256*K3**4 + 296*K3**2*K6 - 2992*K3**2 - 928*K4**2 - 400*K5**2 - 242*K6**2 - 28*K7**2 - 12*K8**2 + 4090 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1147'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4643', 'vk6.4920', 'vk6.6081', 'vk6.6580', 'vk6.8104', 'vk6.8498', 'vk6.9488', 'vk6.9853', 'vk6.20630', 'vk6.22057', 'vk6.28116', 'vk6.29557', 'vk6.39540', 'vk6.41763', 'vk6.46151', 'vk6.47793', 'vk6.48675', 'vk6.48864', 'vk6.49415', 'vk6.49650', 'vk6.50689', 'vk6.50876', 'vk6.51166', 'vk6.51381', 'vk6.57530', 'vk6.58718', 'vk6.62226', 'vk6.63172', 'vk6.67032', 'vk6.67905', 'vk6.69661', 'vk6.70342'] |
| 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 | O1O2O3O4U1U3U2O5O6U5U4U6 |
| R3 orbit | {'O1O2O3O4U1U3U2O5O6U5U4U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U1U6O5O6U3U2U4 |
| Gauss code of K* | O1O2O3U4U5O4O6O5U1U3U2U6 |
| Gauss code of -K* | O1O2O3U1U3O4O5O6U2U5U4U6 |
| 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 0 0 2 -1 2],[ 3 0 2 1 3 0 1],[ 0 -2 0 0 2 0 1],[ 0 -1 0 0 1 0 1],[-2 -3 -2 -1 0 0 2],[ 1 0 0 0 0 0 1],[-2 -1 -1 -1 -2 -1 0]] |
| Primitive based matrix | [[ 0 2 2 0 0 -1 -3],[-2 0 2 -1 -2 0 -3],[-2 -2 0 -1 -1 -1 -1],[ 0 1 1 0 0 0 -1],[ 0 2 1 0 0 0 -2],[ 1 0 1 0 0 0 0],[ 3 3 1 1 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,0,0,1,3,-2,1,2,0,3,1,1,1,1,0,0,1,0,2,0] |
| Phi over symmetry | [-3,-1,0,0,2,2,0,1,2,1,3,0,0,1,0,0,1,1,1,2,-2] |
| Phi of -K | [-3,-1,0,0,2,2,2,1,2,2,4,1,1,3,2,0,0,1,1,1,-2] |
| Phi of K* | [-2,-2,0,0,1,3,-2,1,1,2,4,0,1,3,2,0,1,1,1,2,2] |
| Phi of -K* | [-3,-1,0,0,2,2,0,1,2,1,3,0,0,1,0,0,1,1,1,2,-2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-2t^2+t |
| Normalized Jones-Krushkal polynomial | 4z^2+23z+31 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+23w^2z+31w |
| Inner characteristic polynomial | t^6+27t^4+30t^2 |
| Outer characteristic polynomial | t^7+45t^5+86t^3+5t |
| Flat arrow polynomial | -2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + 2*K4 + 1 |
| 2-strand cable arrow polynomial | -576*K1**3*K3 - 192*K1**2*K2**2 + 288*K1**2*K2*K3**2 + 2976*K1**2*K2 - 2240*K1**2*K3**2 - 96*K1**2*K3*K5 - 48*K1**2*K6**2 - 4932*K1**2 + 64*K1*K2**3*K3 - 1120*K1*K2**2*K3 + 288*K1*K2*K3**3 - 288*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 6200*K1*K2*K3 - 32*K1*K2*K5*K6 - 64*K1*K3**2*K5 + 2832*K1*K3*K4 + 192*K1*K4*K5 + 200*K1*K5*K6 + 80*K1*K6*K7 - 64*K2**4 - 32*K2**3*K6 - 880*K2**2*K3**2 - 8*K2**2*K4**2 + 568*K2**2*K4 - 16*K2**2*K6**2 - 3614*K2**2 - 128*K2*K3**2*K4 + 1032*K2*K3*K5 + 224*K2*K4*K6 + 16*K2*K5*K7 + 32*K2*K6*K8 - 256*K3**4 + 296*K3**2*K6 - 2992*K3**2 - 928*K4**2 - 400*K5**2 - 242*K6**2 - 28*K7**2 - 12*K8**2 + 4090 |
| 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}], [{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}, {2, 5}, {4}, {3}, {1}], [{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}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {2, 3}, {1}]] |
| If K is slice | False |