| Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,2,2,4,0,0,0,1,1,2,2,1,0,-2] |
| Flat knots (up to 7 crossings) with same phi are :['6.660'] |
| Arrow polynomial of the knot is: 8*K1**3 - 4*K1**2 - 6*K1*K2 - 3*K1 + 2*K2 + K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.205', '6.660', '6.775', '6.820'] |
| Outer characteristic polynomial of the knot is: t^7+60t^5+52t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.660'] |
| 2-strand cable arrow polynomial of the knot is: -1216*K1**4*K2**2 + 2272*K1**4*K2 - 2672*K1**4 + 544*K1**3*K2*K3 - 672*K1**3*K3 - 1088*K1**2*K2**4 + 4512*K1**2*K2**3 + 288*K1**2*K2**2*K4 - 11728*K1**2*K2**2 - 992*K1**2*K2*K4 + 9960*K1**2*K2 - 16*K1**2*K3**2 - 80*K1**2*K4**2 - 5144*K1**2 + 1984*K1*K2**3*K3 - 2752*K1*K2**2*K3 - 224*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 8992*K1*K2*K3 + 880*K1*K3*K4 + 128*K1*K4*K5 - 192*K2**6 + 352*K2**4*K4 - 3328*K2**4 - 1104*K2**2*K3**2 - 152*K2**2*K4**2 + 2768*K2**2*K4 - 3038*K2**2 + 376*K2*K3*K5 + 16*K2*K4*K6 - 1892*K3**2 - 716*K4**2 - 36*K5**2 - 2*K6**2 + 4290 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.660'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4723', 'vk6.5042', 'vk6.6250', 'vk6.6700', 'vk6.8224', 'vk6.8666', 'vk6.9608', 'vk6.9933', 'vk6.20294', 'vk6.21629', 'vk6.27586', 'vk6.29140', 'vk6.39008', 'vk6.41258', 'vk6.45772', 'vk6.47451', 'vk6.48755', 'vk6.48956', 'vk6.49554', 'vk6.49770', 'vk6.50769', 'vk6.50975', 'vk6.51248', 'vk6.51455', 'vk6.57153', 'vk6.58339', 'vk6.61775', 'vk6.62896', 'vk6.66774', 'vk6.67652', 'vk6.69418', 'vk6.70142'] |
| 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 | O1O2O3O4U2O5U3U4O6U1U6U5 |
| R3 orbit | {'O1O2O3O4U2O5U3U4O6U1U6U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U6U4O6U1U2O5U3 |
| Gauss code of K* | O1O2O3U1U4U5U6O4U3O5O6U2 |
| Gauss code of -K* | O1O2O3U2O4O5U1O6U4U5U6U3 |
| 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 1 3 1],[ 2 0 -2 0 2 4 1],[ 2 2 0 1 2 2 0],[ 1 0 -1 0 1 2 0],[-1 -2 -2 -1 0 1 0],[-3 -4 -2 -2 -1 0 0],[-1 -1 0 0 0 0 0]] |
| Primitive based matrix | [[ 0 3 1 1 -1 -2 -2],[-3 0 0 -1 -2 -2 -4],[-1 0 0 0 0 0 -1],[-1 1 0 0 -1 -2 -2],[ 1 2 0 1 0 -1 0],[ 2 2 0 2 1 0 2],[ 2 4 1 2 0 -2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,-1,1,2,2,0,1,2,2,4,0,0,0,1,1,2,2,1,0,-2] |
| Phi over symmetry | [-3,-1,-1,1,2,2,0,1,2,2,4,0,0,0,1,1,2,2,1,0,-2] |
| Phi of -K | [-2,-2,-1,1,1,3,-2,0,1,3,3,1,1,2,1,1,2,2,0,1,2] |
| Phi of K* | [-3,-1,-1,1,2,2,1,2,2,1,3,0,1,1,1,2,2,3,1,0,-2] |
| Phi of -K* | [-2,-2,-1,1,1,3,-2,0,1,2,4,1,0,2,2,0,1,2,0,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+2t^2-t |
| Normalized Jones-Krushkal polynomial | 8z^2+29z+27 |
| Enhanced Jones-Krushkal polynomial | 8w^3z^2+29w^2z+27w |
| Inner characteristic polynomial | t^6+40t^4+20t^2+1 |
| Outer characteristic polynomial | t^7+60t^5+52t^3+4t |
| Flat arrow polynomial | 8*K1**3 - 4*K1**2 - 6*K1*K2 - 3*K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | -1216*K1**4*K2**2 + 2272*K1**4*K2 - 2672*K1**4 + 544*K1**3*K2*K3 - 672*K1**3*K3 - 1088*K1**2*K2**4 + 4512*K1**2*K2**3 + 288*K1**2*K2**2*K4 - 11728*K1**2*K2**2 - 992*K1**2*K2*K4 + 9960*K1**2*K2 - 16*K1**2*K3**2 - 80*K1**2*K4**2 - 5144*K1**2 + 1984*K1*K2**3*K3 - 2752*K1*K2**2*K3 - 224*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 8992*K1*K2*K3 + 880*K1*K3*K4 + 128*K1*K4*K5 - 192*K2**6 + 352*K2**4*K4 - 3328*K2**4 - 1104*K2**2*K3**2 - 152*K2**2*K4**2 + 2768*K2**2*K4 - 3038*K2**2 + 376*K2*K3*K5 + 16*K2*K4*K6 - 1892*K3**2 - 716*K4**2 - 36*K5**2 - 2*K6**2 + 4290 |
| 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}], [{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}], [{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}, {4}, {2, 3}, {1}]] |
| If K is slice | False |