| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,2,2,3,0,0,2,1,0,1,0,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1516'] |
| 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+24t^5+37t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1516'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 352*K1**4*K2 - 1424*K1**4 + 352*K1**3*K2*K3 - 544*K1**3*K3 + 128*K1**2*K2**3 - 1344*K1**2*K2**2 - 160*K1**2*K2*K4 + 3376*K1**2*K2 - 464*K1**2*K3**2 - 2224*K1**2 - 96*K1*K2**2*K3 + 2624*K1*K2*K3 + 464*K1*K3*K4 - 56*K2**4 + 112*K2**2*K4 - 1728*K2**2 - 912*K3**2 - 146*K4**2 + 1816 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1516'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4415', 'vk6.4512', 'vk6.4842', 'vk6.5187', 'vk6.5801', 'vk6.5930', 'vk6.6413', 'vk6.6417', 'vk6.6846', 'vk6.7975', 'vk6.8370', 'vk6.8382', 'vk6.8799', 'vk6.9288', 'vk6.9409', 'vk6.9737', 'vk6.17884', 'vk6.17949', 'vk6.18285', 'vk6.18622', 'vk6.24391', 'vk6.25173', 'vk6.30039', 'vk6.30102', 'vk6.36903', 'vk6.37363', 'vk6.39837', 'vk6.39843', 'vk6.43826', 'vk6.44120', 'vk6.44445', 'vk6.46396', 'vk6.46405', 'vk6.47974', 'vk6.47984', 'vk6.48611', 'vk6.49081', 'vk6.49908', 'vk6.50596', 'vk6.51119'] |
| 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 | O1O2O3U2O4U3U1O5O6U5U4U6 |
| R3 orbit | {'O1O2U1O3O4U2U3O5O6U5U4U6', 'O1O2O3U2O4U3U1O5O6U5U4U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3U4U5U6O4O6U3U1O5U2 |
| Gauss code of K* | O1O2O3U4U5U6O5U2O6O4U1U3 |
| Gauss code of -K* | O1O2O3U1U3O4O5U2O6U5U6U4 |
| 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 0 1 -1 2],[ 1 0 -1 1 2 0 1],[ 1 1 0 1 1 0 0],[ 0 -1 -1 0 1 0 1],[-1 -2 -1 -1 0 0 2],[ 1 0 0 0 0 0 1],[-2 -1 0 -1 -2 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 -2 -1 0 -1 -1],[-1 2 0 -1 -1 0 -2],[ 0 1 1 0 -1 0 -1],[ 1 0 1 1 0 0 1],[ 1 1 0 0 0 0 0],[ 1 1 2 1 -1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,2,1,0,1,1,1,1,0,2,1,0,1,0,-1,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,1,2,2,3,0,0,2,1,0,1,0,0,-1,0] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,0,0,1,3,0,0,0,2,1,2,2,0,1,-1] |
| Phi of K* | [-2,-1,0,1,1,1,-1,1,2,2,3,0,0,2,1,0,1,0,0,-1,0] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,0,1,2,1,0,1,1,0,0,0,1,1,1,2] |
| 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+16t^4+12t^2 |
| Outer characteristic polynomial | t^7+24t^5+37t^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 - 1424*K1**4 + 352*K1**3*K2*K3 - 544*K1**3*K3 + 128*K1**2*K2**3 - 1344*K1**2*K2**2 - 160*K1**2*K2*K4 + 3376*K1**2*K2 - 464*K1**2*K3**2 - 2224*K1**2 - 96*K1*K2**2*K3 + 2624*K1*K2*K3 + 464*K1*K3*K4 - 56*K2**4 + 112*K2**2*K4 - 1728*K2**2 - 912*K3**2 - 146*K4**2 + 1816 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}]] |
| If K is slice | False |