| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,0,2,2,2,0,1,1,2,0,1,1,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1838'] |
| Arrow polynomial of the knot is: -6*K1**2 - 4*K1*K2 + 2*K1 + 3*K2 + 2*K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.239', '6.428', '6.470', '6.556', '6.700', '6.910', '6.962', '6.1006', '6.1013', '6.1038', '6.1207', '6.1224', '6.1225', '6.1269', '6.1270', '6.1308', '6.1319', '6.1320', '6.1323', '6.1485', '6.1551', '6.1579', '6.1581', '6.1660', '6.1672', '6.1679', '6.1711', '6.1719', '6.1732', '6.1745', '6.1748', '6.1827', '6.1836', '6.1838', '6.1850', '6.1866'] |
| Outer characteristic polynomial of the knot is: t^7+24t^5+49t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1838'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 192*K1**4*K2 - 624*K1**4 + 64*K1**3*K2*K3 - 352*K1**3*K3 + 512*K1**2*K2**3 - 4176*K1**2*K2**2 - 448*K1**2*K2*K4 + 6136*K1**2*K2 - 144*K1**2*K3**2 - 5096*K1**2 + 416*K1*K2**3*K3 - 768*K1*K2**2*K3 - 128*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 6088*K1*K2*K3 + 672*K1*K3*K4 + 40*K1*K4*K5 - 1704*K2**4 - 752*K2**2*K3**2 - 48*K2**2*K4**2 + 1752*K2**2*K4 - 3284*K2**2 + 512*K2*K3*K5 + 32*K2*K4*K6 - 1852*K3**2 - 506*K4**2 - 68*K5**2 - 4*K6**2 + 3760 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1838'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73751', 'vk6.73760', 'vk6.73891', 'vk6.73897', 'vk6.75697', 'vk6.75708', 'vk6.75893', 'vk6.75899', 'vk6.78680', 'vk6.78696', 'vk6.78880', 'vk6.78892', 'vk6.80306', 'vk6.80320', 'vk6.80433', 'vk6.80442', 'vk6.81697', 'vk6.81706', 'vk6.81708', 'vk6.81801', 'vk6.82200', 'vk6.82456', 'vk6.82463', 'vk6.82466', 'vk6.82470', 'vk6.84426', 'vk6.84438', 'vk6.84444', 'vk6.87783', 'vk6.88117', 'vk6.88397', 'vk6.89634'] |
| 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 | O1O2O3U1U4O5U3O6O4U2U5U6 |
| R3 orbit | {'O1O2O3U1U4O5U3O6O4U2U5U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5U2O6O4U1O5U6U3 |
| Gauss code of K* | O1O2O3U4U1U5O4O6U2O5U3U6 |
| Gauss code of -K* | O1O2O3U4U1O5U2O4O6U5U3U6 |
| 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 -1 1 1 0 1],[ 2 0 2 1 1 2 1],[ 1 -2 0 1 0 1 1],[-1 -1 -1 0 -1 0 0],[-1 -1 0 1 0 0 0],[ 0 -2 -1 0 0 0 1],[-1 -1 -1 0 0 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 0 0 -1],[-1 -1 0 0 0 -1 -1],[-1 0 0 0 -1 -1 -1],[ 0 0 0 1 0 -1 -2],[ 1 0 1 1 1 0 -2],[ 2 1 1 1 2 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,0,0,1,0,0,1,1,1,1,1,1,2,2] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,0,2,2,2,0,1,1,2,0,1,1,0,0,1] |
| Phi of -K | [-2,-1,0,1,1,1,-1,0,2,2,2,0,1,1,2,0,1,1,0,0,1] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,1,1,2,0,1,2,2,0,1,2,0,0,-1] |
| Phi of -K* | [-2,-1,0,1,1,1,2,2,1,1,1,1,0,1,1,0,0,1,1,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 5z^2+22z+25 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+7w^3z^2-2w^3z+24w^2z+25w |
| Inner characteristic polynomial | t^6+16t^4+16t^2 |
| Outer characteristic polynomial | t^7+24t^5+49t^3+8t |
| Flat arrow polynomial | -6*K1**2 - 4*K1*K2 + 2*K1 + 3*K2 + 2*K3 + 4 |
| 2-strand cable arrow polynomial | -64*K1**4*K2**2 + 192*K1**4*K2 - 624*K1**4 + 64*K1**3*K2*K3 - 352*K1**3*K3 + 512*K1**2*K2**3 - 4176*K1**2*K2**2 - 448*K1**2*K2*K4 + 6136*K1**2*K2 - 144*K1**2*K3**2 - 5096*K1**2 + 416*K1*K2**3*K3 - 768*K1*K2**2*K3 - 128*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 6088*K1*K2*K3 + 672*K1*K3*K4 + 40*K1*K4*K5 - 1704*K2**4 - 752*K2**2*K3**2 - 48*K2**2*K4**2 + 1752*K2**2*K4 - 3284*K2**2 + 512*K2*K3*K5 + 32*K2*K4*K6 - 1852*K3**2 - 506*K4**2 - 68*K5**2 - 4*K6**2 + 3760 |
| 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}, {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}, {4, 5}, {3}, {2}, {1}], [{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 |