| Min(phi) over symmetries of the knot is: [-3,0,1,2,1,1,4,0,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1037'] |
| Arrow polynomial of the knot is: -6*K1**2 - 6*K1*K2 + 3*K1 + 3*K2 + 3*K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.458', '6.601', '6.611', '6.995', '6.1026', '6.1037'] |
| Outer characteristic polynomial of the knot is: t^5+33t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1037'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**6 - 128*K1**4*K2**2 + 608*K1**4*K2 - 1248*K1**4 + 128*K1**3*K2*K3 - 784*K1**2*K2**2 + 1728*K1**2*K2 - 352*K1**2*K3**2 - 128*K1**2*K4**2 - 964*K1**2 + 1128*K1*K2*K3 + 616*K1*K3*K4 + 216*K1*K4*K5 + 16*K1*K5*K6 - 56*K2**4 - 48*K2**2*K3**2 - 24*K2**2*K4**2 + 192*K2**2*K4 - 1082*K2**2 + 184*K2*K3*K5 + 56*K2*K4*K6 + 16*K3**2*K6 - 632*K3**2 - 370*K4**2 - 164*K5**2 - 38*K6**2 + 1320 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1037'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4144', 'vk6.4175', 'vk6.5386', 'vk6.5417', 'vk6.5482', 'vk6.5593', 'vk6.7512', 'vk6.7678', 'vk6.9017', 'vk6.9048', 'vk6.11177', 'vk6.12265', 'vk6.12372', 'vk6.12449', 'vk6.12480', 'vk6.13358', 'vk6.13583', 'vk6.13614', 'vk6.14257', 'vk6.14704', 'vk6.14732', 'vk6.15191', 'vk6.15864', 'vk6.15892', 'vk6.26200', 'vk6.26645', 'vk6.30850', 'vk6.30881', 'vk6.32038', 'vk6.32069', 'vk6.33084', 'vk6.33115', 'vk6.38145', 'vk6.38176', 'vk6.44806', 'vk6.44921', 'vk6.49226', 'vk6.49335', 'vk6.52763', 'vk6.53530'] |
| 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 | O1O2O3O4U5U2O5U3O6U1U6U4 |
| R3 orbit | {'O1O2O3O4U3U5U2O5O6U1U6U4', 'O1O2O3O4U5U2O5U3O6U1U6U4'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U1U5U4O5U2O6U3U6 |
| Gauss code of K* | O1O2O3U1U4U5U3O6O4U6O5U2 |
| Gauss code of -K* | O1O2O3U2O4U5O6O5U1U4U6U3 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 2 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -2 -1 0 3 -1 1],[ 2 0 0 1 4 1 1],[ 1 0 0 0 1 1 0],[ 0 -1 0 0 1 0 0],[-3 -4 -1 -1 0 -3 0],[ 1 -1 -1 0 3 0 1],[-1 -1 0 0 0 -1 0]] |
| Primitive based matrix | [[ 0 3 0 -1 -2],[-3 0 -1 -1 -4],[ 0 1 0 0 -1],[ 1 1 0 0 0],[ 2 4 1 0 0]] |
| If based matrix primitive | False |
| Phi of primitive based matrix | [-3,0,1,2,1,1,4,0,1,0] |
| Phi over symmetry | [-3,0,1,2,1,1,4,0,1,0] |
| Phi of -K | [-2,-1,0,3,1,1,1,1,3,2] |
| Phi of K* | [-3,0,1,2,2,3,1,1,1,1] |
| Phi of -K* | [-2,-1,0,3,0,1,4,0,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 11z+23 |
| Enhanced Jones-Krushkal polynomial | 11w^2z+23w |
| Inner characteristic polynomial | t^4+19t^2+1 |
| Outer characteristic polynomial | t^5+33t^3+8t |
| Flat arrow polynomial | -6*K1**2 - 6*K1*K2 + 3*K1 + 3*K2 + 3*K3 + 4 |
| 2-strand cable arrow polynomial | -192*K1**6 - 128*K1**4*K2**2 + 608*K1**4*K2 - 1248*K1**4 + 128*K1**3*K2*K3 - 784*K1**2*K2**2 + 1728*K1**2*K2 - 352*K1**2*K3**2 - 128*K1**2*K4**2 - 964*K1**2 + 1128*K1*K2*K3 + 616*K1*K3*K4 + 216*K1*K4*K5 + 16*K1*K5*K6 - 56*K2**4 - 48*K2**2*K3**2 - 24*K2**2*K4**2 + 192*K2**2*K4 - 1082*K2**2 + 184*K2*K3*K5 + 56*K2*K4*K6 + 16*K3**2*K6 - 632*K3**2 - 370*K4**2 - 164*K5**2 - 38*K6**2 + 1320 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {5}, {1, 4}, {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}, {4}, {3}, {1, 2}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {3}, {2}]] |
| If K is slice | False |