| Min(phi) over symmetries of the knot is: [-3,0,0,0,1,2,1,1,2,0,3,-1,-1,1,-1,0,0,1,0,2,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1044'] |
| 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+61t^5+297t^3+9t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1044'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4 + 32*K1**3*K2*K3 - 64*K1**3*K3 - 80*K1**2*K2**2 - 32*K1**2*K2*K4 + 776*K1**2*K2 - 1744*K1**2*K3**2 - 2660*K1**2 - 224*K1*K2**2*K3 + 32*K1*K2*K3**3 + 4208*K1*K2*K3 - 64*K1*K3**2*K5 + 1936*K1*K3*K4 + 32*K1*K4*K5 + 48*K1*K5*K6 - 8*K2**4 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 264*K2**2*K4 - 2138*K2**2 + 184*K2*K3*K5 + 16*K2*K4*K6 - 16*K3**4 + 80*K3**2*K6 - 2212*K3**2 - 626*K4**2 - 104*K5**2 - 54*K6**2 + 2552 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1044'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4655', 'vk6.4944', 'vk6.6105', 'vk6.6594', 'vk6.8118', 'vk6.8522', 'vk6.9508', 'vk6.9865', 'vk6.20372', 'vk6.21715', 'vk6.27684', 'vk6.29230', 'vk6.39124', 'vk6.41380', 'vk6.45868', 'vk6.47531', 'vk6.48695', 'vk6.48900', 'vk6.49455', 'vk6.49676', 'vk6.50715', 'vk6.50916', 'vk6.51198', 'vk6.51401', 'vk6.57241', 'vk6.58468', 'vk6.61867', 'vk6.63004', 'vk6.66868', 'vk6.67738', 'vk6.69492', 'vk6.70216'] |
| 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 | O1O2O3O4U5U6O5U1O6U3U2U4 |
| R3 orbit | {'O1O2O3O4U5U6O5U1O6U3U2U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U3U2O5U4O6U5U6 |
| Gauss code of K* | O1O2O3U4U2U1U3O5O6U5O4U6 |
| Gauss code of -K* | O1O2O3U4O5U6O4O6U1U3U2U5 |
| 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 3 -1 0],[ 2 0 1 0 2 2 3],[ 0 -1 0 0 2 -1 1],[ 0 0 0 0 1 -1 1],[-3 -2 -2 -1 0 -4 -2],[ 1 -2 1 1 4 0 0],[ 0 -3 -1 -1 2 0 0]] |
| Primitive based matrix | [[ 0 3 0 0 0 -1 -2],[-3 0 -1 -2 -2 -4 -2],[ 0 1 0 1 0 -1 0],[ 0 2 -1 0 -1 0 -3],[ 0 2 0 1 0 -1 -1],[ 1 4 1 0 1 0 -2],[ 2 2 0 3 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,0,0,0,1,2,1,2,2,4,2,-1,0,1,0,1,0,3,1,1,2] |
| Phi over symmetry | [-3,0,0,0,1,2,1,1,2,0,3,-1,-1,1,-1,0,0,1,0,2,-1] |
| Phi of -K | [-2,-1,0,0,0,3,-1,-1,1,2,3,1,0,0,0,1,1,1,0,1,2] |
| Phi of K* | [-3,0,0,0,1,2,1,1,2,0,3,-1,-1,1,-1,0,0,1,0,2,-1] |
| Phi of -K* | [-2,-1,0,0,0,3,2,0,1,3,2,1,1,0,4,0,1,1,1,2,2] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 4z^2+21z+27 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2-4w^3z+25w^2z+27w |
| Inner characteristic polynomial | t^6+47t^4+216t^2+1 |
| Outer characteristic polynomial | t^7+61t^5+297t^3+9t |
| Flat arrow polynomial | -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -64*K1**4 + 32*K1**3*K2*K3 - 64*K1**3*K3 - 80*K1**2*K2**2 - 32*K1**2*K2*K4 + 776*K1**2*K2 - 1744*K1**2*K3**2 - 2660*K1**2 - 224*K1*K2**2*K3 + 32*K1*K2*K3**3 + 4208*K1*K2*K3 - 64*K1*K3**2*K5 + 1936*K1*K3*K4 + 32*K1*K4*K5 + 48*K1*K5*K6 - 8*K2**4 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 264*K2**2*K4 - 2138*K2**2 + 184*K2*K3*K5 + 16*K2*K4*K6 - 16*K3**4 + 80*K3**2*K6 - 2212*K3**2 - 626*K4**2 - 104*K5**2 - 54*K6**2 + 2552 |
| 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}, {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}, {3, 5}, {4}, {2}, {1}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {4, 5}, {3}, {2}, {1}], [{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}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {1, 3}, {2}]] |
| If K is slice | False |