| Min(phi) over symmetries of the knot is: [-1,-1,-1,1,1,1,-1,-1,1,1,1,-1,1,1,1,0,1,1,-1,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.414', '6.1611', '7.14293', '7.20044', '7.25964'] |
| 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+19t^5+23t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.414', '6.1611', '7.20044'] |
| 2-strand cable arrow polynomial of the knot is: -512*K1**6 + 2176*K1**4*K2 - 8832*K1**4 + 64*K1**3*K2*K3 - 864*K1**3*K3 - 3264*K1**2*K2**2 - 32*K1**2*K2*K4 + 11520*K1**2*K2 - 128*K1**2*K3**2 - 2520*K1**2 + 2976*K1*K2*K3 + 112*K1*K3*K4 - 64*K2**4 + 64*K2**2*K4 - 3640*K2**2 - 696*K3**2 - 40*K4**2 + 3678 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1611'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.18879', 'vk6.18880', 'vk6.18881', 'vk6.18892', 'vk6.18896', 'vk6.18900', 'vk6.18956', 'vk6.18957', 'vk6.18959', 'vk6.18968', 'vk6.18972', 'vk6.18976', 'vk6.25582', 'vk6.25583', 'vk6.25584', 'vk6.25595', 'vk6.25599', 'vk6.25603', 'vk6.37606', 'vk6.37607', 'vk6.37608', 'vk6.37623', 'vk6.37627', 'vk6.37635', 'vk6.56421', 'vk6.56422', 'vk6.56461', 'vk6.56462', 'vk6.56464', 'vk6.56465', 'vk6.56473', 'vk6.56477'] |
| 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 | O1O2O3U2O4U5U1O5U3O6U4U6 |
| R3 orbit | {'O1O2O3U2O4U5U1O5U3O6U4U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5O4U1O6U3U6O5U2 |
| Gauss code of K* | O1O2U1O3U4O5O4U2U6U3O6U5 |
| Gauss code of -K* | O1O2U1O3U4O5O4U2O6U3U6U5 |
| 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 1 1 -1 1],[ 1 0 0 1 1 1 1],[ 1 0 0 1 1 1 0],[-1 -1 -1 0 1 -1 1],[-1 -1 -1 -1 0 -1 1],[ 1 -1 -1 1 1 0 1],[-1 -1 0 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 -1 -1 -1],[-1 0 1 1 -1 -1 -1],[-1 -1 0 1 -1 -1 -1],[-1 -1 -1 0 0 -1 -1],[ 1 1 1 0 0 1 0],[ 1 1 1 1 -1 0 -1],[ 1 1 1 1 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,1,1,1,-1,-1,1,1,1,-1,1,1,1,0,1,1,-1,0,1] |
| Phi over symmetry | [-1,-1,-1,1,1,1,-1,-1,1,1,1,-1,1,1,1,0,1,1,-1,0,1] |
| Phi of -K | [-1,-1,-1,1,1,1,-1,0,1,1,1,1,1,1,1,1,1,2,-1,-1,-1] |
| Phi of K* | [-1,-1,-1,1,1,1,-1,-1,1,1,2,-1,1,1,1,1,1,1,-1,-1,0] |
| Phi of -K* | [-1,-1,-1,1,1,1,-1,-1,1,1,1,0,0,1,1,1,1,1,-1,-1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | z^2+22z+41 |
| Enhanced Jones-Krushkal polynomial | w^3z^2+22w^2z+41w |
| Inner characteristic polynomial | t^6+13t^4+9t^2+1 |
| Outer characteristic polynomial | t^7+19t^5+23t^3+5t |
| Flat arrow polynomial | -8*K1**2 + 4*K2 + 5 |
| 2-strand cable arrow polynomial | -512*K1**6 + 2176*K1**4*K2 - 8832*K1**4 + 64*K1**3*K2*K3 - 864*K1**3*K3 - 3264*K1**2*K2**2 - 32*K1**2*K2*K4 + 11520*K1**2*K2 - 128*K1**2*K3**2 - 2520*K1**2 + 2976*K1*K2*K3 + 112*K1*K3*K4 - 64*K2**4 + 64*K2**2*K4 - 3640*K2**2 - 696*K3**2 - 40*K4**2 + 3678 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}]] |
| If K is slice | True |