| Min(phi) over symmetries of the knot is: [-1,0,0,0,1,0,0,1,1,-1,-1,1,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.2018', '7.42182', '7.45710'] |
| 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^6+7t^4+12t^2+1 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.2018', '7.27332', '7.42182', '7.45710'] |
| 2-strand cable arrow polynomial of the knot is: 544*K1**4*K2 - 7136*K1**4 - 352*K1**3*K3 - 2144*K1**2*K2**2 + 10392*K1**2*K2 - 2896*K1**2 + 2072*K1*K2*K3 - 128*K2**4 + 128*K2**2*K4 - 3384*K2**2 - 480*K3**2 - 32*K4**2 + 3414 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.2018'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73292', 'vk6.73304', 'vk6.73433', 'vk6.73445', 'vk6.74084', 'vk6.74093', 'vk6.74655', 'vk6.74662', 'vk6.75437', 'vk6.75449', 'vk6.76118', 'vk6.76127', 'vk6.78173', 'vk6.78177', 'vk6.78403', 'vk6.78407', 'vk6.79094', 'vk6.79103', 'vk6.79998', 'vk6.80002', 'vk6.80149', 'vk6.80153', 'vk6.80598', 'vk6.80607', 'vk6.83810', 'vk6.83817', 'vk6.85122', 'vk6.85136', 'vk6.86600', 'vk6.86622', 'vk6.87372', 'vk6.87400'] |
| 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 | O1O2U3O4U2O5U6U1O3O6U4U5 |
| R3 orbit | {'O1O2U3O4U2O5U6U1O3O6U4U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2U3U4O5O6U2U5O3U1O4U6 |
| Gauss code of K* | O1O2U3U1O4O5U2U6O3U4O6U5 |
| Gauss code of -K* | O1O2U3O4U5O6U4U1O3O5U2U6 |
| 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 0 0 -1 0 1 0],[ 0 0 0 -1 0 0 1],[ 0 0 0 0 0 0 1],[ 1 1 0 0 1 1 0],[ 0 0 0 -1 0 1 0],[-1 0 0 -1 -1 0 -1],[ 0 -1 -1 0 0 1 0]] |
| Primitive based matrix | [[ 0 1 0 0 0 -1],[-1 0 0 0 -1 -1],[ 0 0 0 0 1 0],[ 0 0 0 0 1 -1],[ 0 1 -1 -1 0 0],[ 1 1 0 1 0 0]] |
| If based matrix primitive | False |
| Phi of primitive based matrix | [-1,0,0,0,1,0,0,1,1,0,-1,0,-1,1,0] |
| Phi over symmetry | [-1,0,0,0,1,0,0,1,1,-1,-1,1,0,0,0] |
| Phi of -K | [-1,0,0,0,1,0,1,1,1,-1,0,1,1,0,1] |
| Phi of K* | [-1,0,0,0,1,0,1,1,1,-1,-1,1,0,0,1] |
| Phi of -K* | [-1,0,0,0,1,0,0,1,1,-1,-1,1,0,0,0] |
| 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^5+5t^3+6t |
| Outer characteristic polynomial | t^6+7t^4+12t^2+1 |
| Flat arrow polynomial | -8*K1**2 + 4*K2 + 5 |
| 2-strand cable arrow polynomial | 544*K1**4*K2 - 7136*K1**4 - 352*K1**3*K3 - 2144*K1**2*K2**2 + 10392*K1**2*K2 - 2896*K1**2 + 2072*K1*K2*K3 - 128*K2**4 + 128*K2**2*K4 - 3384*K2**2 - 480*K3**2 - 32*K4**2 + 3414 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {4}, {2, 3}, {1}], [{6}, {3, 5}, {4}, {1, 2}]] |
| If K is slice | False |