| Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,1,1,2,3,1,1,1,2,1,0,1,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.754'] |
| 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+43t^5+29t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.754'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 352*K1**4*K2 - 528*K1**4 + 128*K1**2*K2**3 - 2464*K1**2*K2**2 + 2776*K1**2*K2 - 48*K1**2*K3**2 - 1616*K1**2 + 1816*K1*K2*K3 + 48*K1*K3*K4 - 56*K2**4 + 24*K2**2*K4 - 1064*K2**2 - 336*K3**2 - 18*K4**2 + 1112 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.754'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11276', 'vk6.11354', 'vk6.12539', 'vk6.12650', 'vk6.18363', 'vk6.18703', 'vk6.24809', 'vk6.25268', 'vk6.30959', 'vk6.31083', 'vk6.31208', 'vk6.31549', 'vk6.32136', 'vk6.32255', 'vk6.32376', 'vk6.32787', 'vk6.36999', 'vk6.37449', 'vk6.39621', 'vk6.41862', 'vk6.44177', 'vk6.44498', 'vk6.46229', 'vk6.47836', 'vk6.52044', 'vk6.52483', 'vk6.52885', 'vk6.53365', 'vk6.56363', 'vk6.57611', 'vk6.61001', 'vk6.62275', 'vk6.63663', 'vk6.63708', 'vk6.64093', 'vk6.64138', 'vk6.65797', 'vk6.66053', 'vk6.68795', 'vk6.69005'] |
| 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 | O1O2O3O4U2O5U4U6U1O6U5U3 |
| R3 orbit | {'O1O2O3O4U2U3O5U6U1O6U4U5', 'O1O2O3O4U2O5U4U6U1O6U5U3'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U2U5O6U4U6U1O5U3 |
| Gauss code of K* | O1O2O3U2O4O5U3U6U5U1O6U4 |
| Gauss code of -K* | O1O2O3U4O5U3U6U5U1O6O4U2 |
| 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 -2 2 0 2 -1],[ 1 0 -1 2 1 1 1],[ 2 1 0 2 1 1 2],[-2 -2 -2 0 -1 1 -2],[ 0 -1 -1 1 0 1 0],[-2 -1 -1 -1 -1 0 -2],[ 1 -1 -2 2 0 2 0]] |
| Primitive based matrix | [[ 0 2 2 0 -1 -1 -2],[-2 0 1 -1 -2 -2 -2],[-2 -1 0 -1 -1 -2 -1],[ 0 1 1 0 -1 0 -1],[ 1 2 1 1 0 1 -1],[ 1 2 2 0 -1 0 -2],[ 2 2 1 1 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,0,1,1,2,-1,1,2,2,2,1,1,2,1,1,0,1,-1,1,2] |
| Phi over symmetry | [-2,-2,0,1,1,2,-1,1,1,2,3,1,1,1,2,1,0,1,-1,-1,0] |
| Phi of -K | [-2,-1,-1,0,2,2,-1,0,1,2,3,1,1,1,1,0,1,2,1,1,-1] |
| Phi of K* | [-2,-2,0,1,1,2,-1,1,1,2,3,1,1,1,2,1,0,1,-1,-1,0] |
| Phi of -K* | [-2,-1,-1,0,2,2,1,2,1,1,2,1,1,1,2,0,2,2,1,1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+16w^2z+21w |
| Inner characteristic polynomial | t^6+29t^4+8t^2 |
| Outer characteristic polynomial | t^7+43t^5+29t^3+3t |
| Flat arrow polynomial | -6*K1**2 + 3*K2 + 4 |
| 2-strand cable arrow polynomial | -128*K1**4*K2**2 + 352*K1**4*K2 - 528*K1**4 + 128*K1**2*K2**3 - 2464*K1**2*K2**2 + 2776*K1**2*K2 - 48*K1**2*K3**2 - 1616*K1**2 + 1816*K1*K2*K3 + 48*K1*K3*K4 - 56*K2**4 + 24*K2**2*K4 - 1064*K2**2 - 336*K3**2 - 18*K4**2 + 1112 |
| 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}], [{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}], [{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}, {3}, {1, 2}]] |
| If K is slice | False |