| Min(phi) over symmetries of the knot is: [-3,0,0,0,1,2,0,1,2,2,4,1,1,-1,0,0,-1,1,0,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.641'] |
| 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+51t^5+89t^3+19t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.641'] |
| 2-strand cable arrow polynomial of the knot is: -16*K1**4 - 2304*K1**2*K2**2 + 2528*K1**2*K2 - 80*K1**2*K3**2 - 2628*K1**2 + 192*K1*K2**3*K3 - 288*K1*K2**2*K3 - 32*K1*K2**2*K5 + 3952*K1*K2*K3 + 192*K1*K3*K4 + 32*K1*K4*K5 - 808*K2**4 - 304*K2**2*K3**2 - 8*K2**2*K4**2 + 688*K2**2*K4 - 1846*K2**2 + 240*K2*K3*K5 + 8*K2*K4*K6 - 1408*K3**2 - 178*K4**2 - 76*K5**2 - 2*K6**2 + 2144 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.641'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72631', 'vk6.72634', 'vk6.72786', 'vk6.72793', 'vk6.73099', 'vk6.73108', 'vk6.73174', 'vk6.73179', 'vk6.73806', 'vk6.73813', 'vk6.73953', 'vk6.73955', 'vk6.73960', 'vk6.73965', 'vk6.75765', 'vk6.75768', 'vk6.75775', 'vk6.75786', 'vk6.77877', 'vk6.77933', 'vk6.77991', 'vk6.78018', 'vk6.78767', 'vk6.78774', 'vk6.80367', 'vk6.80369', 'vk6.80373', 'vk6.80375', 'vk6.81786', 'vk6.87807', 'vk6.89147', 'vk6.89336'] |
| 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 | O1O2O3O4U1O5U4U3O6U5U2U6 |
| R3 orbit | {'O1O2O3O4U1O5U4U3O6U5U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U3U6O5U2U1O6U4 |
| Gauss code of K* | O1O2O3U4U2U5U6O4U1O6O5U3 |
| Gauss code of -K* | O1O2O3U1O4O5U3O6U5U4U2U6 |
| 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 0 1 2],[ 3 0 3 2 1 2 1],[ 0 -3 0 -1 -1 2 2],[ 0 -2 1 0 0 2 1],[ 0 -1 1 0 0 1 1],[-1 -2 -2 -2 -1 0 1],[-2 -1 -2 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 0 -3],[-2 0 -1 -1 -1 -2 -1],[-1 1 0 -1 -2 -2 -2],[ 0 1 1 0 0 1 -1],[ 0 1 2 0 0 1 -2],[ 0 2 2 -1 -1 0 -3],[ 3 1 2 1 2 3 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,0,3,1,1,1,2,1,1,2,2,2,0,-1,1,-1,2,3] |
| Phi over symmetry | [-3,0,0,0,1,2,0,1,2,2,4,1,1,-1,0,0,-1,1,0,1,0] |
| Phi of -K | [-3,0,0,0,1,2,0,1,2,2,4,1,1,-1,0,0,-1,1,0,1,0] |
| Phi of K* | [-2,-1,0,0,0,3,0,0,1,1,4,-1,-1,0,2,-1,-1,0,0,1,2] |
| Phi of -K* | [-3,0,0,0,1,2,1,2,3,2,1,0,1,1,1,1,2,1,2,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 5z+11 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+4w^3z^2-16w^3z+21w^2z+11w |
| Inner characteristic polynomial | t^6+37t^4+16t^2+1 |
| Outer characteristic polynomial | t^7+51t^5+89t^3+19t |
| Flat arrow polynomial | -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -16*K1**4 - 2304*K1**2*K2**2 + 2528*K1**2*K2 - 80*K1**2*K3**2 - 2628*K1**2 + 192*K1*K2**3*K3 - 288*K1*K2**2*K3 - 32*K1*K2**2*K5 + 3952*K1*K2*K3 + 192*K1*K3*K4 + 32*K1*K4*K5 - 808*K2**4 - 304*K2**2*K3**2 - 8*K2**2*K4**2 + 688*K2**2*K4 - 1846*K2**2 + 240*K2*K3*K5 + 8*K2*K4*K6 - 1408*K3**2 - 178*K4**2 - 76*K5**2 - 2*K6**2 + 2144 |
| 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}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}]] |
| If K is slice | False |