| Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,0,2,2,3,-1,-1,1,1,-1,1,0,1,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1566'] |
| 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+31t^5+166t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1566'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 + 192*K1**4*K2 - 1296*K1**4 + 160*K1**3*K2*K3 - 320*K1**3*K3 - 1440*K1**2*K2**2 + 3752*K1**2*K2 - 208*K1**2*K3**2 - 2340*K1**2 - 160*K1*K2**2*K3 + 2232*K1*K2*K3 + 208*K1*K3*K4 - 56*K2**4 + 96*K2**2*K4 - 1768*K2**2 - 716*K3**2 - 66*K4**2 + 1792 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1566'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16563', 'vk6.16656', 'vk6.18145', 'vk6.18479', 'vk6.22966', 'vk6.23087', 'vk6.24604', 'vk6.25015', 'vk6.34955', 'vk6.35076', 'vk6.35378', 'vk6.35799', 'vk6.36735', 'vk6.37152', 'vk6.39420', 'vk6.41611', 'vk6.42528', 'vk6.42639', 'vk6.42855', 'vk6.43134', 'vk6.44007', 'vk6.44317', 'vk6.46000', 'vk6.47674', 'vk6.54794', 'vk6.55346', 'vk6.56257', 'vk6.57418', 'vk6.59226', 'vk6.59787', 'vk6.60861', 'vk6.62089', 'vk6.64768', 'vk6.64833', 'vk6.65616', 'vk6.65921', 'vk6.68070', 'vk6.68135', 'vk6.68691', 'vk6.68900'] |
| 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 | O1O2O3U4O5U6U2O4O6U3U1U5 |
| R3 orbit | {'O1O2O3U4O5U3U6O4U2O6U1U5', 'O1O2O3U4O5U6U2O4O6U3U1U5'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3U4U3U1O5O6U2U5O4U6 |
| Gauss code of K* | O1O2O3U2U4U1O5U3O6O4U5U6 |
| Gauss code of -K* | O1O2O3U4U5O6O4U1O5U3U6U2 |
| 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 0 0 -1 2 0],[ 1 0 1 1 -1 2 1],[ 0 -1 0 -1 0 0 1],[ 0 -1 1 0 -1 0 1],[ 1 1 0 1 0 3 0],[-2 -2 0 0 -3 0 -2],[ 0 -1 -1 -1 0 2 0]] |
| Primitive based matrix | [[ 0 2 0 0 0 -1 -1],[-2 0 0 0 -2 -2 -3],[ 0 0 0 1 1 -1 -1],[ 0 0 -1 0 1 -1 0],[ 0 2 -1 -1 0 -1 0],[ 1 2 1 1 1 0 -1],[ 1 3 1 0 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,0,0,0,1,1,0,0,2,2,3,-1,-1,1,1,-1,1,0,1,0,1] |
| Phi over symmetry | [-2,0,0,0,1,1,0,0,2,2,3,-1,-1,1,1,-1,1,0,1,0,1] |
| Phi of -K | [-1,-1,0,0,0,2,-1,0,1,1,0,0,0,0,1,-1,-1,2,-1,2,0] |
| Phi of K* | [-2,0,0,0,1,1,0,2,2,0,1,-1,-1,1,0,-1,1,0,0,0,1] |
| Phi of -K* | [-1,-1,0,0,0,2,-1,1,1,1,2,0,0,1,3,-1,-1,2,-1,0,0] |
| 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+25t^4+121t^2+1 |
| Outer characteristic polynomial | t^7+31t^5+166t^3+4t |
| Flat arrow polynomial | -6*K1**2 + 3*K2 + 4 |
| 2-strand cable arrow polynomial | -64*K1**6 + 192*K1**4*K2 - 1296*K1**4 + 160*K1**3*K2*K3 - 320*K1**3*K3 - 1440*K1**2*K2**2 + 3752*K1**2*K2 - 208*K1**2*K3**2 - 2340*K1**2 - 160*K1*K2**2*K3 + 2232*K1*K2*K3 + 208*K1*K3*K4 - 56*K2**4 + 96*K2**2*K4 - 1768*K2**2 - 716*K3**2 - 66*K4**2 + 1792 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {2, 4}, {1, 3}]] |
| If K is slice | False |