| Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,0,-1,0,1,2,0,0,1,1,-1,0,0,1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1958'] |
| Arrow polynomial of the knot is: -8*K1**2 + 4*K2 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.668', '6.711', '6.777', '6.803', '6.828', '6.1015', '6.1032', '6.1055', '6.1082', '6.1132', '6.1264', '6.1288', '6.1333', '6.1391', '6.1395', '6.1396', '6.1400', '6.1404', '6.1405', '6.1419', '6.1471', '6.1473', '6.1536', '6.1563', '6.1611', '6.1618', '6.1623', '6.1627', '6.1629', '6.1631', '6.1695', '6.1700', '6.1731', '6.1740', '6.1767', '6.1773', '6.1790', '6.1792', '6.1796', '6.1848', '6.1899', '6.1901', '6.1937', '6.1954', '6.1955', '6.1958', '6.1964', '6.1975', '6.1997', '6.1998', '6.1999', '6.2002', '6.2003', '6.2004', '6.2005', '6.2007', '6.2008', '6.2009', '6.2010', '6.2011', '6.2013', '6.2018', '6.2019', '6.2021', '6.2034', '6.2039', '6.2043', '6.2046', '6.2050', '6.2051', '6.2057', '6.2063'] |
| Outer characteristic polynomial of the knot is: t^7+18t^5+26t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1958', '7.43499'] |
| 2-strand cable arrow polynomial of the knot is: -320*K1**6 - 320*K1**4*K2**2 + 3168*K1**4*K2 - 6624*K1**4 + 448*K1**3*K2*K3 - 1632*K1**3*K3 + 320*K1**2*K2**3 - 5504*K1**2*K2**2 - 320*K1**2*K2*K4 + 11008*K1**2*K2 - 160*K1**2*K3**2 - 4132*K1**2 - 224*K1*K2**2*K3 + 5248*K1*K2*K3 + 264*K1*K3*K4 - 224*K2**4 + 328*K2**2*K4 - 3976*K2**2 - 1220*K3**2 - 140*K4**2 + 4010 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1958'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4075', 'vk6.4108', 'vk6.5317', 'vk6.5350', 'vk6.7439', 'vk6.7468', 'vk6.8940', 'vk6.8973', 'vk6.10107', 'vk6.10272', 'vk6.10297', 'vk6.14535', 'vk6.15287', 'vk6.15414', 'vk6.15753', 'vk6.16170', 'vk6.29855', 'vk6.29888', 'vk6.33921', 'vk6.34004', 'vk6.34210', 'vk6.34388', 'vk6.48473', 'vk6.49180', 'vk6.50219', 'vk6.50250', 'vk6.51583', 'vk6.53980', 'vk6.54037', 'vk6.54173', 'vk6.54485', 'vk6.63304'] |
| 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 | O1O2U1O3O4U3U5O6U4O5U2U6 |
| R3 orbit | {'O1O2U1O3O4U3U5O6U4O5U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2U3O4O3U5U4O6U1O5U6U2 |
| Gauss code of K* | O1O2U3O4U2O5O3U6U5O6U1U4 |
| Gauss code of -K* | O1O2U3O4U1O3O5U4U5O6U2U6 |
| 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 -1 1 0 1],[ 1 0 1 0 0 1 1],[ 0 -1 0 -1 2 -1 1],[ 1 0 1 0 1 0 1],[-1 0 -2 -1 0 -1 0],[ 0 -1 1 0 1 0 1],[-1 -1 -1 -1 0 -1 0]] |
| Primitive based matrix | [[ 0 1 1 0 0 -1 -1],[-1 0 0 -1 -1 -1 -1],[-1 0 0 -1 -2 0 -1],[ 0 1 1 0 1 -1 0],[ 0 1 2 -1 0 -1 -1],[ 1 1 0 1 1 0 0],[ 1 1 1 0 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,0,0,1,1,0,1,1,1,1,1,2,0,1,-1,1,0,1,1,0] |
| Phi over symmetry | [-1,-1,0,0,1,1,0,-1,0,1,2,0,0,1,1,-1,0,0,1,0,0] |
| Phi of -K | [-1,-1,0,0,1,1,0,0,0,1,2,0,1,1,1,1,0,-1,0,0,0] |
| Phi of K* | [-1,-1,0,0,1,1,0,-1,0,1,2,0,0,1,1,-1,0,0,1,0,0] |
| Phi of -K* | [-1,-1,0,0,1,1,0,0,1,1,1,1,1,0,1,1,1,1,2,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 3z^2+24z+37 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+24w^2z+37w |
| Inner characteristic polynomial | t^6+14t^4+14t^2+1 |
| Outer characteristic polynomial | t^7+18t^5+26t^3+4t |
| Flat arrow polynomial | -8*K1**2 + 4*K2 + 5 |
| 2-strand cable arrow polynomial | -320*K1**6 - 320*K1**4*K2**2 + 3168*K1**4*K2 - 6624*K1**4 + 448*K1**3*K2*K3 - 1632*K1**3*K3 + 320*K1**2*K2**3 - 5504*K1**2*K2**2 - 320*K1**2*K2*K4 + 11008*K1**2*K2 - 160*K1**2*K3**2 - 4132*K1**2 - 224*K1*K2**2*K3 + 5248*K1*K2*K3 + 264*K1*K3*K4 - 224*K2**4 + 328*K2**2*K4 - 3976*K2**2 - 1220*K3**2 - 140*K4**2 + 4010 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}]] |
| If K is slice | False |