| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,0,0,1,2,3,-1,0,0,2,1,0,1,-1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1695'] |
| 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+34t^5+109t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1695'] |
| 2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 1152*K1**4*K2 - 5344*K1**4 + 192*K1**3*K2*K3 - 384*K1**3*K3 + 1536*K1**2*K2**3 - 8672*K1**2*K2**2 - 256*K1**2*K2*K4 + 11552*K1**2*K2 - 3736*K1**2 - 768*K1*K2**2*K3 + 6064*K1*K2*K3 - 1568*K2**4 + 1056*K2**2*K4 - 3072*K2**2 - 952*K3**2 - 72*K4**2 + 3654 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1695'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81559', 'vk6.81635', 'vk6.81828', 'vk6.82048', 'vk6.82221', 'vk6.82333', 'vk6.82538', 'vk6.82995', 'vk6.83131', 'vk6.83571', 'vk6.83934', 'vk6.84073', 'vk6.84524', 'vk6.84886', 'vk6.85909', 'vk6.86415', 'vk6.86465', 'vk6.88830', 'vk6.89772', 'vk6.89881'] |
| The R3 orbit of minmal crossing diagrams contains: |
| The diagrammatic symmetry type of this knot is -. |
| The reverse -K is |
| The 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 | O1O2O3U4O5U6U1U2O4O6U3U5 |
| R3 orbit | {'O1O2O3U4O5U6U1U2O4O6U3U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U1O5O6U2U3U5O4U6 |
| Gauss code of K* | O1O2O3U4U1O5O6U2U3U5O4U6 |
| Gauss code of -K* | Same |
| Diagrammatic symmetry type | - |
| 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 1 -2 2 -1],[ 1 0 1 0 0 1 1],[-1 -1 0 -1 -1 0 0],[-1 0 1 0 -2 0 0],[ 2 0 1 2 0 3 0],[-2 -1 0 0 -3 0 -2],[ 1 -1 0 0 0 2 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 0 0 -1 -2 -3],[-1 0 0 1 0 0 -2],[-1 0 -1 0 -1 0 -1],[ 1 1 0 1 0 1 0],[ 1 2 0 0 -1 0 0],[ 2 3 2 1 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,0,0,1,2,3,-1,0,0,2,1,0,1,-1,0,0] |
| Phi over symmetry | [-2,-1,-1,1,1,2,0,0,1,2,3,-1,0,0,2,1,0,1,-1,0,0] |
| Phi of -K | [-2,-1,-1,1,1,2,1,1,1,2,1,-1,2,1,2,2,2,1,-1,1,1] |
| Phi of K* | [-2,-1,-1,1,1,2,1,1,1,2,1,-1,2,1,2,2,2,1,-1,1,1] |
| Phi of -K* | [-2,-1,-1,1,1,2,0,0,1,2,3,-1,0,0,2,1,0,1,-1,0,0] |
| Symmetry type of based matrix | - |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 4z^2+24z+33 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+24w^2z+33w |
| Inner characteristic polynomial | t^6+22t^4+65t^2+1 |
| Outer characteristic polynomial | t^7+34t^5+109t^3+7t |
| Flat arrow polynomial | -8*K1**2 + 4*K2 + 5 |
| 2-strand cable arrow polynomial | -384*K1**4*K2**2 + 1152*K1**4*K2 - 5344*K1**4 + 192*K1**3*K2*K3 - 384*K1**3*K3 + 1536*K1**2*K2**3 - 8672*K1**2*K2**2 - 256*K1**2*K2*K4 + 11552*K1**2*K2 - 3736*K1**2 - 768*K1*K2**2*K3 + 6064*K1*K2*K3 - 1568*K2**4 + 1056*K2**2*K4 - 3072*K2**2 - 952*K3**2 - 72*K4**2 + 3654 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{3, 6}, {4, 5}, {1, 2}]] |
| If K is slice | True |