| Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,1,1,1,2,1,1,1,2,1,1,2,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.910'] |
| Arrow polynomial of the knot is: -6*K1**2 - 4*K1*K2 + 2*K1 + 3*K2 + 2*K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.239', '6.428', '6.470', '6.556', '6.700', '6.910', '6.962', '6.1006', '6.1013', '6.1038', '6.1207', '6.1224', '6.1225', '6.1269', '6.1270', '6.1308', '6.1319', '6.1320', '6.1323', '6.1485', '6.1551', '6.1579', '6.1581', '6.1660', '6.1672', '6.1679', '6.1711', '6.1719', '6.1732', '6.1745', '6.1748', '6.1827', '6.1836', '6.1838', '6.1850', '6.1866'] |
| Outer characteristic polynomial of the knot is: t^7+47t^5+53t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.910'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 + 224*K1**4*K2 - 864*K1**4 + 64*K1**3*K2*K3 - 736*K1**3*K3 - 832*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 96*K1**2*K2*K4 + 3728*K1**2*K2 - 480*K1**2*K3**2 - 48*K1**2*K4**2 - 3076*K1**2 - 384*K1*K2**2*K3 - 192*K1*K2*K3*K4 + 2968*K1*K2*K3 + 712*K1*K3*K4 + 72*K1*K4*K5 - 56*K2**4 - 192*K2**2*K3**2 - 48*K2**2*K4**2 + 360*K2**2*K4 - 2164*K2**2 + 168*K2*K3*K5 + 32*K2*K4*K6 - 1072*K3**2 - 278*K4**2 - 28*K5**2 - 4*K6**2 + 2156 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.910'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16568', 'vk6.16659', 'vk6.18132', 'vk6.18468', 'vk6.22971', 'vk6.23090', 'vk6.24591', 'vk6.25004', 'vk6.34968', 'vk6.35087', 'vk6.35391', 'vk6.35810', 'vk6.36730', 'vk6.37149', 'vk6.39391', 'vk6.41584', 'vk6.42541', 'vk6.42650', 'vk6.42868', 'vk6.43145', 'vk6.44002', 'vk6.44314', 'vk6.45971', 'vk6.47647', 'vk6.54815', 'vk6.55365', 'vk6.56230', 'vk6.57413', 'vk6.59247', 'vk6.59806', 'vk6.60834', 'vk6.62084', 'vk6.64797', 'vk6.64860', 'vk6.65595', 'vk6.65902', 'vk6.68099', 'vk6.68162', 'vk6.68670', 'vk6.68881'] |
| 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 | O1O2O3O4U2U5O6O5U4U1U3U6 |
| R3 orbit | {'O1O2O3U1O4U5O6O5U2U4U3U6', 'O1O2O3O4U2U5O6O5U4U1U3U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U2U4U1O6O5U6U3 |
| Gauss code of K* | O1O2O3O4U2U5U3U1O5O6U4U6 |
| Gauss code of -K* | O1O2O3O4U5U1O5O6U4U2U6U3 |
| 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 1 0 1 2],[ 2 0 -1 2 1 2 2],[ 2 1 0 2 1 2 2],[-1 -2 -2 0 0 -1 1],[ 0 -1 -1 0 0 0 0],[-1 -2 -2 1 0 0 2],[-2 -2 -2 -1 0 -2 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -2 -2],[-2 0 -1 -2 0 -2 -2],[-1 1 0 -1 0 -2 -2],[-1 2 1 0 0 -2 -2],[ 0 0 0 0 0 -1 -1],[ 2 2 2 2 1 0 1],[ 2 2 2 2 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,2,2,1,2,0,2,2,1,0,2,2,0,2,2,1,1,-1] |
| Phi over symmetry | [-2,-2,0,1,1,2,-1,1,1,1,2,1,1,1,2,1,1,2,-1,-1,0] |
| Phi of -K | [-2,-2,0,1,1,2,-1,1,1,1,2,1,1,1,2,1,1,2,-1,-1,0] |
| Phi of K* | [-2,-1,-1,0,2,2,-1,0,2,2,2,1,1,1,1,1,1,1,1,1,-1] |
| Phi of -K* | [-2,-2,0,1,1,2,-1,1,2,2,2,1,2,2,2,0,0,0,-1,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 13z+27 |
| Enhanced Jones-Krushkal polynomial | 13w^2z+27w |
| Inner characteristic polynomial | t^6+33t^4+18t^2 |
| Outer characteristic polynomial | t^7+47t^5+53t^3+3t |
| Flat arrow polynomial | -6*K1**2 - 4*K1*K2 + 2*K1 + 3*K2 + 2*K3 + 4 |
| 2-strand cable arrow polynomial | -64*K1**6 + 224*K1**4*K2 - 864*K1**4 + 64*K1**3*K2*K3 - 736*K1**3*K3 - 832*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 96*K1**2*K2*K4 + 3728*K1**2*K2 - 480*K1**2*K3**2 - 48*K1**2*K4**2 - 3076*K1**2 - 384*K1*K2**2*K3 - 192*K1*K2*K3*K4 + 2968*K1*K2*K3 + 712*K1*K3*K4 + 72*K1*K4*K5 - 56*K2**4 - 192*K2**2*K3**2 - 48*K2**2*K4**2 + 360*K2**2*K4 - 2164*K2**2 + 168*K2*K3*K5 + 32*K2*K4*K6 - 1072*K3**2 - 278*K4**2 - 28*K5**2 - 4*K6**2 + 2156 |
| 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}], [{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}, {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 |