| Min(phi) over symmetries of the knot is: [-3,0,0,0,1,2,0,1,2,3,3,1,1,0,0,0,1,0,1,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.710'] |
| Arrow polynomial of the knot is: -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.217', '6.219', '6.304', '6.349', '6.390', '6.400', '6.416', '6.515', '6.518', '6.530', '6.582', '6.616', '6.629', '6.641', '6.645', '6.702', '6.710', '6.715', '6.729', '6.733', '6.734', '6.802', '6.840', '6.845', '6.854', '6.860', '6.900', '6.905', '6.921', '6.924', '6.979', '6.980', '6.996', '6.1044', '6.1067', '6.1086', '6.1100', '6.1139', '6.1145', '6.1149', '6.1167', '6.1169', '6.1183', '6.1314'] |
| Outer characteristic polynomial of the knot is: t^7+43t^5+106t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.710'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4 + 96*K1**3*K2*K3 - 64*K1**3*K3 - 1184*K1**2*K2**2 - 32*K1**2*K2*K4 + 1728*K1**2*K2 - 64*K1**2*K3**2 - 1520*K1**2 - 288*K1*K2**2*K3 + 1872*K1*K2*K3 + 168*K1*K3*K4 - 120*K2**4 - 96*K2**2*K3**2 - 8*K2**2*K4**2 + 224*K2**2*K4 - 1102*K2**2 + 64*K2*K3*K5 + 8*K2*K4*K6 - 608*K3**2 - 82*K4**2 - 8*K5**2 - 2*K6**2 + 1080 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.710'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4231', 'vk6.4312', 'vk6.5500', 'vk6.5618', 'vk6.7607', 'vk6.7698', 'vk6.9100', 'vk6.9181', 'vk6.18368', 'vk6.18708', 'vk6.24817', 'vk6.25276', 'vk6.37005', 'vk6.37455', 'vk6.44182', 'vk6.44503', 'vk6.48551', 'vk6.48608', 'vk6.49256', 'vk6.49376', 'vk6.50348', 'vk6.50405', 'vk6.51081', 'vk6.51114', 'vk6.56149', 'vk6.56378', 'vk6.60674', 'vk6.61025', 'vk6.65817', 'vk6.66071', 'vk6.68810', 'vk6.69020', 'vk6.83725', 'vk6.83858', 'vk6.85056', 'vk6.85335', 'vk6.86669', 'vk6.86990', 'vk6.87406', 'vk6.89521'] |
| 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 | O1O2O3O4U5O6U4U1O5U3U2U6 |
| R3 orbit | {'O1O2O3O4U5U3O5U6U2U1O6U4', 'O1O2O3O4U5O6U4U1O5U3U2U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U3U2O6U4U1O5U6 |
| Gauss code of K* | O1O2O3U4U2U1U5O6U3O5O4U6 |
| Gauss code of -K* | O1O2O3U4O5O6U1O4U6U3U2U5 |
| 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 0 0 0 -1 3],[ 2 0 1 0 0 1 3],[ 0 -1 0 0 1 -1 2],[ 0 0 0 0 1 -1 1],[ 0 0 -1 -1 0 0 0],[ 1 -1 1 1 0 0 3],[-3 -3 -2 -1 0 -3 0]] |
| Primitive based matrix | [[ 0 3 0 0 0 -1 -2],[-3 0 0 -1 -2 -3 -3],[ 0 0 0 -1 -1 0 0],[ 0 1 1 0 0 -1 0],[ 0 2 1 0 0 -1 -1],[ 1 3 0 1 1 0 -1],[ 2 3 0 0 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,0,0,0,1,2,0,1,2,3,3,1,1,0,0,0,1,0,1,1,1] |
| Phi over symmetry | [-3,0,0,0,1,2,0,1,2,3,3,1,1,0,0,0,1,0,1,1,1] |
| Phi of -K | [-2,-1,0,0,0,3,0,1,2,2,2,0,0,1,1,0,-1,1,-1,2,3] |
| Phi of K* | [-3,0,0,0,1,2,1,2,3,1,2,0,1,0,1,1,0,2,1,2,0] |
| Phi of -K* | [-2,-1,0,0,0,3,1,0,0,1,3,0,1,1,3,-1,-1,0,0,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 5z^2+18z+17 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+18w^2z+17w |
| Inner characteristic polynomial | t^6+29t^4+63t^2+4 |
| Outer characteristic polynomial | t^7+43t^5+106t^3+7t |
| Flat arrow polynomial | -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -64*K1**4 + 96*K1**3*K2*K3 - 64*K1**3*K3 - 1184*K1**2*K2**2 - 32*K1**2*K2*K4 + 1728*K1**2*K2 - 64*K1**2*K3**2 - 1520*K1**2 - 288*K1*K2**2*K3 + 1872*K1*K2*K3 + 168*K1*K3*K4 - 120*K2**4 - 96*K2**2*K3**2 - 8*K2**2*K4**2 + 224*K2**2*K4 - 1102*K2**2 + 64*K2*K3*K5 + 8*K2*K4*K6 - 608*K3**2 - 82*K4**2 - 8*K5**2 - 2*K6**2 + 1080 |
| 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}], [{1, 6}, {5}, {4}, {3}, {2}], [{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}], [{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}, {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}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {2, 3}, {1}]] |
| If K is slice | False |