| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,2,2,0,1,1,2,0,0,0,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1570', '7.37777'] |
| 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+22t^5+29t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1570', '7.37777'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**6 - 128*K1**4*K2**2 + 672*K1**4*K2 - 1856*K1**4 + 64*K1**3*K2*K3 - 160*K1**3*K3 + 608*K1**2*K2**3 - 2896*K1**2*K2**2 - 96*K1**2*K2*K4 + 4000*K1**2*K2 - 1324*K1**2 - 480*K1*K2**2*K3 + 1968*K1*K2*K3 + 56*K1*K3*K4 - 600*K2**4 + 488*K2**2*K4 - 1112*K2**2 - 300*K3**2 - 54*K4**2 + 1276 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1570'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11052', 'vk6.11132', 'vk6.12214', 'vk6.12323', 'vk6.16417', 'vk6.19230', 'vk6.19325', 'vk6.19523', 'vk6.19618', 'vk6.22716', 'vk6.22817', 'vk6.26040', 'vk6.26093', 'vk6.26424', 'vk6.26515', 'vk6.30629', 'vk6.30726', 'vk6.31933', 'vk6.34770', 'vk6.35487', 'vk6.35939', 'vk6.38112', 'vk6.38129', 'vk6.38872', 'vk6.41070', 'vk6.42385', 'vk6.42916', 'vk6.43216', 'vk6.44633', 'vk6.44755', 'vk6.45633', 'vk6.51845', 'vk6.52807', 'vk6.55195', 'vk6.58220', 'vk6.59578', 'vk6.62779', 'vk6.64716', 'vk6.66289', 'vk6.66293'] |
| 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 | O1O2O3U4O5U1U5O6O4U2U3U6 |
| R3 orbit | {'O1O2U3O4O5U1U5O3O6U2U6U4', 'O1O2O3U4O5U1U5O6O4U2U3U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3U4U1U2O5O4U6U3O6U5 |
| Gauss code of K* | O1O2O3U4U1U2O5U6O4O6U3U5 |
| Gauss code of -K* | O1O2O3U4U1O5O6U5O4U2U3U6 |
| 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 0 1 1],[ 2 0 1 2 1 1 1],[ 1 -1 0 1 1 0 1],[-1 -2 -1 0 -1 0 0],[ 0 -1 -1 1 0 1 1],[-1 -1 0 0 -1 0 0],[-1 -1 -1 0 -1 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 0 0 -1 0 -1],[-1 0 0 0 -1 -1 -1],[-1 0 0 0 -1 -1 -2],[ 0 1 1 1 0 -1 -1],[ 1 0 1 1 1 0 -1],[ 2 1 1 2 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,0,0,1,0,1,0,1,1,1,1,1,2,1,1,1] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,1,1,2,2,0,1,1,2,0,0,0,0,0,0] |
| Phi of -K | [-2,-1,0,1,1,1,0,1,1,2,2,0,1,1,2,0,0,0,0,0,0] |
| Phi of K* | [-1,-1,-1,0,1,2,0,0,0,1,1,0,0,1,2,0,2,2,0,1,0] |
| Phi of -K* | [-2,-1,0,1,1,1,1,1,1,1,2,1,0,1,1,1,1,1,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 2z^2+15z+23 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2+15w^2z+23w |
| Inner characteristic polynomial | t^6+14t^4+16t^2+1 |
| Outer characteristic polynomial | t^7+22t^5+29t^3+4t |
| Flat arrow polynomial | -6*K1**2 + 3*K2 + 4 |
| 2-strand cable arrow polynomial | -128*K1**6 - 128*K1**4*K2**2 + 672*K1**4*K2 - 1856*K1**4 + 64*K1**3*K2*K3 - 160*K1**3*K3 + 608*K1**2*K2**3 - 2896*K1**2*K2**2 - 96*K1**2*K2*K4 + 4000*K1**2*K2 - 1324*K1**2 - 480*K1*K2**2*K3 + 1968*K1*K2*K3 + 56*K1*K3*K4 - 600*K2**4 + 488*K2**2*K4 - 1112*K2**2 - 300*K3**2 - 54*K4**2 + 1276 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{6}, {2, 5}, {3, 4}, {1}]] |
| If K is slice | False |