| Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,2,1,2,4,1,1,1,2,1,1,2,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.444'] |
| Arrow polynomial of the knot is: -6*K1**2 - 2*K1*K2 + K1 + 3*K2 + K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.323', '6.380', '6.444', '6.472', '6.523', '6.579', '6.592', '6.595', '6.609', '6.614', '6.620', '6.644', '6.648', '6.669', '6.671', '6.681', '6.693', '6.724', '6.725', '6.757', '6.766', '6.785', '6.786', '6.797', '6.798', '6.816', '6.833', '6.972', '6.978', '6.1056', '6.1064', '6.1066', '6.1087', '6.1094', '6.1273', '6.1277', '6.1282', '6.1295', '6.1300', '6.1313', '6.1344', '6.1353', '6.1354'] |
| Outer characteristic polynomial of the knot is: t^7+67t^5+136t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.444'] |
| 2-strand cable arrow polynomial of the knot is: -544*K1**4 + 32*K1**3*K2*K3 - 288*K1**3*K3 - 784*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 64*K1**2*K2*K4 + 3144*K1**2*K2 - 448*K1**2*K3**2 - 32*K1**2*K3*K5 - 2812*K1**2 - 544*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 2632*K1*K2*K3 + 680*K1*K3*K4 + 40*K1*K4*K5 - 56*K2**4 - 192*K2**2*K3**2 - 8*K2**2*K4**2 + 304*K2**2*K4 - 2022*K2**2 + 200*K2*K3*K5 + 16*K2*K4*K6 + 24*K3**2*K6 - 1044*K3**2 - 242*K4**2 - 64*K5**2 - 18*K6**2 + 2024 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.444'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16507', 'vk6.16600', 'vk6.18076', 'vk6.18414', 'vk6.22938', 'vk6.23035', 'vk6.23496', 'vk6.23835', 'vk6.24527', 'vk6.24946', 'vk6.35024', 'vk6.35645', 'vk6.36666', 'vk6.37090', 'vk6.39443', 'vk6.41644', 'vk6.42484', 'vk6.42597', 'vk6.43946', 'vk6.44263', 'vk6.46031', 'vk6.47699', 'vk6.54750', 'vk6.54847', 'vk6.56194', 'vk6.57449', 'vk6.59214', 'vk6.59279', 'vk6.59662', 'vk6.60010', 'vk6.60793', 'vk6.62124', 'vk6.64829', 'vk6.65062', 'vk6.65550', 'vk6.65862', 'vk6.68066', 'vk6.68131', 'vk6.68632', 'vk6.68847'] |
| 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 | O1O2O3O4O5U6U2O6U4U1U5U3 |
| R3 orbit | {'O1O2O3O4O5U6U2O6U4U1U5U3', 'O1O2O3O4U5U1O5U6U2U4O6U3'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U3U1U5U2O6U4U6 |
| Gauss code of K* | O1O2O3O4U2U5U4U1U3O6O5U6 |
| Gauss code of -K* | O1O2O3O4U5O6O5U2U4U1U6U3 |
| 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 2 0 3 -1],[ 2 0 0 3 1 3 1],[ 2 0 0 2 0 1 2],[-2 -3 -2 0 -1 1 -2],[ 0 -1 0 1 0 1 0],[-3 -3 -1 -1 -1 0 -3],[ 1 -1 -2 2 0 3 0]] |
| Primitive based matrix | [[ 0 3 2 0 -1 -2 -2],[-3 0 -1 -1 -3 -1 -3],[-2 1 0 -1 -2 -2 -3],[ 0 1 1 0 0 0 -1],[ 1 3 2 0 0 -2 -1],[ 2 1 2 0 2 0 0],[ 2 3 3 1 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,0,1,2,2,1,1,3,1,3,1,2,2,3,0,0,1,2,1,0] |
| Phi over symmetry | [-3,-2,0,1,2,2,0,2,1,2,4,1,1,1,2,1,1,2,0,-1,0] |
| Phi of -K | [-2,-2,-1,0,2,3,0,-1,2,2,4,0,1,1,2,1,1,1,1,2,0] |
| Phi of K* | [-3,-2,0,1,2,2,0,2,1,2,4,1,1,1,2,1,1,2,0,-1,0] |
| Phi of -K* | [-2,-2,-1,0,2,3,0,1,1,3,3,2,0,2,1,0,2,3,1,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 13z+27 |
| Enhanced Jones-Krushkal polynomial | 13w^2z+27w |
| Inner characteristic polynomial | t^6+45t^4+83t^2+1 |
| Outer characteristic polynomial | t^7+67t^5+136t^3+4t |
| Flat arrow polynomial | -6*K1**2 - 2*K1*K2 + K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -544*K1**4 + 32*K1**3*K2*K3 - 288*K1**3*K3 - 784*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 64*K1**2*K2*K4 + 3144*K1**2*K2 - 448*K1**2*K3**2 - 32*K1**2*K3*K5 - 2812*K1**2 - 544*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 2632*K1*K2*K3 + 680*K1*K3*K4 + 40*K1*K4*K5 - 56*K2**4 - 192*K2**2*K3**2 - 8*K2**2*K4**2 + 304*K2**2*K4 - 2022*K2**2 + 200*K2*K3*K5 + 16*K2*K4*K6 + 24*K3**2*K6 - 1044*K3**2 - 242*K4**2 - 64*K5**2 - 18*K6**2 + 2024 |
| 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}], [{2, 6}, {5}, {4}, {3}, {1}], [{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}, {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}, {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}, {4}, {1, 3}, {2}]] |
| If K is slice | False |