| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,1,3,1,0,0,1,0,1,1,0,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1431'] |
| 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+26t^5+20t^3+2t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1431'] |
| 2-strand cable arrow polynomial of the knot is: -640*K1**6 + 1728*K1**4*K2 - 6128*K1**4 + 288*K1**3*K2*K3 - 608*K1**3*K3 - 3552*K1**2*K2**2 - 32*K1**2*K2*K4 + 8848*K1**2*K2 - 272*K1**2*K3**2 - 2512*K1**2 + 3808*K1*K2*K3 + 136*K1*K3*K4 - 120*K2**4 + 152*K2**2*K4 - 3280*K2**2 - 1064*K3**2 - 74*K4**2 + 3320 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1431'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4441', 'vk6.4538', 'vk6.5823', 'vk6.5952', 'vk6.7887', 'vk6.8001', 'vk6.9310', 'vk6.9431', 'vk6.13418', 'vk6.13515', 'vk6.13706', 'vk6.14055', 'vk6.15030', 'vk6.15152', 'vk6.17776', 'vk6.17809', 'vk6.18836', 'vk6.19436', 'vk6.19731', 'vk6.24323', 'vk6.25433', 'vk6.25466', 'vk6.26614', 'vk6.33264', 'vk6.33325', 'vk6.37563', 'vk6.44897', 'vk6.48646', 'vk6.50546', 'vk6.53664', 'vk6.55810', 'vk6.65476'] |
| 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 | O1O2O3U2O4U3O5U1U5O6U4U6 |
| R3 orbit | {'O1O2O3U2O4U3O5U1U5O6U4U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5O4U6U3O6U1O5U2 |
| Gauss code of K* | O1O2U3O4O3U1U5U6O5U4O6U2 |
| Gauss code of -K* | O1O2U1O3O4U3O5U2O6U5U6U4 |
| 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 0 1 1 1],[ 2 0 -1 1 3 1 1],[ 1 1 0 1 1 0 0],[ 0 -1 -1 0 1 0 1],[-1 -3 -1 -1 0 0 1],[-1 -1 0 0 0 0 0],[-1 -1 0 -1 -1 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -1 -1 -3],[-1 -1 0 0 -1 0 -1],[-1 0 0 0 0 0 -1],[ 0 1 1 0 0 -1 -1],[ 1 1 0 0 1 0 1],[ 2 3 1 1 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,1,1,3,0,1,0,1,0,0,1,1,1,-1] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,1,1,1,3,1,0,0,1,0,1,1,0,0,-1] |
| Phi of -K | [-2,-1,0,1,1,1,2,1,0,2,2,0,1,2,2,0,0,1,-1,0,0] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,0,2,2,0,0,1,0,1,2,2,0,1,2] |
| Phi of -K* | [-2,-1,0,1,1,1,-1,1,1,1,3,1,0,0,1,0,1,1,0,0,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 2z^2+23z+39 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2+23w^2z+39w |
| Inner characteristic polynomial | t^6+18t^4+7t^2 |
| Outer characteristic polynomial | t^7+26t^5+20t^3+2t |
| Flat arrow polynomial | -6*K1**2 + 3*K2 + 4 |
| 2-strand cable arrow polynomial | -640*K1**6 + 1728*K1**4*K2 - 6128*K1**4 + 288*K1**3*K2*K3 - 608*K1**3*K3 - 3552*K1**2*K2**2 - 32*K1**2*K2*K4 + 8848*K1**2*K2 - 272*K1**2*K3**2 - 2512*K1**2 + 3808*K1*K2*K3 + 136*K1*K3*K4 - 120*K2**4 + 152*K2**2*K4 - 3280*K2**2 - 1064*K3**2 - 74*K4**2 + 3320 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}]] |
| If K is slice | False |