| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,2,3,0,0,0,1,1,0,0,-1,-1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.691'] |
| Arrow polynomial of the knot is: -6*K1**2 + 3*K2 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.689', '6.691', '6.752', '6.754', '6.1106', '6.1116', '6.1126', '6.1335', '6.1379', '6.1386', '6.1409', '6.1415', '6.1417', '6.1418', '6.1421', '6.1422', '6.1428', '6.1431', '6.1432', '6.1435', '6.1443', '6.1445', '6.1446', '6.1447', '6.1454', '6.1455', '6.1460', '6.1462', '6.1464', '6.1466', '6.1472', '6.1474', '6.1475', '6.1501', '6.1516', '6.1518', '6.1566', '6.1570', '6.1590', '6.1599', '6.1602', '6.1603', '6.1604', '6.1605', '6.1614', '6.1615', '6.1625', '6.1628', '6.1730', '6.1780', '6.1883', '6.1885', '6.1888', '6.1890', '6.1941', '6.1943', '6.1945', '6.1948', '6.1961', '6.1963', '6.1966', '6.1967', '6.1971'] |
| Outer characteristic polynomial of the knot is: t^7+30t^5+25t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.691'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 - 64*K1**4*K2**2 + 704*K1**4*K2 - 2192*K1**4 + 160*K1**3*K2*K3 - 768*K1**3*K3 - 1248*K1**2*K2**2 - 96*K1**2*K2*K4 + 3896*K1**2*K2 - 112*K1**2*K3**2 - 1628*K1**2 + 1768*K1*K2*K3 + 96*K1*K3*K4 - 24*K2**4 + 80*K2**2*K4 - 1440*K2**2 - 452*K3**2 - 46*K4**2 + 1428 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.691'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4086', 'vk6.4117', 'vk6.4243', 'vk6.4322', 'vk6.5324', 'vk6.5355', 'vk6.5518', 'vk6.5540', 'vk6.5637', 'vk6.5659', 'vk6.7477', 'vk6.7706', 'vk6.8947', 'vk6.8978', 'vk6.9108', 'vk6.9187', 'vk6.14546', 'vk6.15290', 'vk6.15419', 'vk6.15766', 'vk6.16181', 'vk6.26280', 'vk6.26725', 'vk6.29846', 'vk6.29877', 'vk6.33932', 'vk6.34213', 'vk6.38219', 'vk6.38232', 'vk6.44992', 'vk6.45009', 'vk6.48565', 'vk6.49169', 'vk6.49276', 'vk6.49282', 'vk6.50243', 'vk6.51588', 'vk6.53967', 'vk6.54472', 'vk6.63308'] |
| 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 | O1O2O3O4U3O5U4U2O6U5U1U6 |
| R3 orbit | {'O1O2O3O4U3O5U4U2O6U5U1U6', 'O1O2O3U2O4O5U3U4O6U5U1U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U4U6O5U3U1O6U2 |
| Gauss code of K* | O1O2O3U2U4U5U6O5U1O6O4U3 |
| Gauss code of -K* | O1O2O3U1O4O5U3O6U5U6U4U2 |
| 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 -1 -1 -1 0 1 2],[ 1 0 -1 -1 0 2 2],[ 1 1 0 -1 1 2 1],[ 1 1 1 0 1 1 0],[ 0 0 -1 -1 0 1 1],[-1 -2 -2 -1 -1 0 1],[-2 -2 -1 0 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 -1 -1 0 -1 -2],[-1 1 0 -1 -1 -2 -2],[ 0 1 1 0 -1 -1 0],[ 1 0 1 1 0 1 1],[ 1 1 2 1 -1 0 1],[ 1 2 2 0 -1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,1,1,0,1,2,1,1,2,2,1,1,0,-1,-1,-1] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,1,1,2,3,0,0,0,1,1,0,0,-1,-1,-1] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,-1,0,1,3,-1,0,0,2,1,0,1,0,1,0] |
| Phi of K* | [-2,-1,0,1,1,1,0,1,1,2,3,0,0,0,1,1,0,0,-1,-1,-1] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,-1,0,2,2,-1,1,2,1,1,1,0,1,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 13z+27 |
| Enhanced Jones-Krushkal polynomial | 13w^2z+27w |
| Inner characteristic polynomial | t^6+22t^4+8t^2 |
| Outer characteristic polynomial | t^7+30t^5+25t^3+3t |
| Flat arrow polynomial | -6*K1**2 + 3*K2 + 4 |
| 2-strand cable arrow polynomial | -64*K1**6 - 64*K1**4*K2**2 + 704*K1**4*K2 - 2192*K1**4 + 160*K1**3*K2*K3 - 768*K1**3*K3 - 1248*K1**2*K2**2 - 96*K1**2*K2*K4 + 3896*K1**2*K2 - 112*K1**2*K3**2 - 1628*K1**2 + 1768*K1*K2*K3 + 96*K1*K3*K4 - 24*K2**4 + 80*K2**2*K4 - 1440*K2**2 - 452*K3**2 - 46*K4**2 + 1428 |
| 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}], [{1, 6}, {5}, {4}, {3}, {2}], [{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}], [{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}, {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}, {3, 4}, {2}, {1}]] |
| If K is slice | False |