| Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,-1,1,2,2,3,0,1,1,2,-1,-1,-1,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1631'] |
| 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+42t^5+35t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1631'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**6 - 384*K1**4*K2**2 + 1312*K1**4*K2 - 3488*K1**4 + 320*K1**3*K2*K3 - 864*K1**3*K3 + 288*K1**2*K2**3 - 2720*K1**2*K2**2 - 192*K1**2*K2*K4 + 6656*K1**2*K2 - 192*K1**2*K3**2 - 2940*K1**2 - 160*K1*K2**2*K3 + 3088*K1*K2*K3 + 264*K1*K3*K4 - 160*K2**4 + 160*K2**2*K4 - 2464*K2**2 - 796*K3**2 - 88*K4**2 + 2550 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1631'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4221', 'vk6.4300', 'vk6.5486', 'vk6.5597', 'vk6.7582', 'vk6.7674', 'vk6.9086', 'vk6.9165', 'vk6.11161', 'vk6.12249', 'vk6.12356', 'vk6.19369', 'vk6.19662', 'vk6.19779', 'vk6.26151', 'vk6.26216', 'vk6.26567', 'vk6.26661', 'vk6.30759', 'vk6.31964', 'vk6.38147', 'vk6.38192', 'vk6.44804', 'vk6.44937', 'vk6.48531', 'vk6.49228', 'vk6.49339', 'vk6.50315', 'vk6.52751', 'vk6.63591', 'vk6.66315', 'vk6.66348'] |
| 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 | O1O2O3U1O4U5U3O6U4U6O5U2 |
| R3 orbit | {'O1O2O3U1O4U5U3O6U4U6O5U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U2O4U5U6O5U1U4O6U3 |
| Gauss code of K* | O1O2U3O4O3U1O5U6U5U2O6U4 |
| Gauss code of -K* | O1O2U1O3O4U2O5U3U6U5O6U4 |
| 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 -2 1 1 1 -2 1],[ 2 0 2 1 1 1 0],[-1 -2 0 0 1 -3 1],[-1 -1 0 0 1 -2 1],[-1 -1 -1 -1 0 -2 1],[ 2 -1 3 2 2 0 1],[-1 0 -1 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 1 -2 -2],[-1 0 1 1 0 -1 -2],[-1 -1 0 1 -1 -1 -2],[-1 -1 -1 0 -1 0 -1],[-1 0 1 1 0 -2 -3],[ 2 1 1 0 2 0 1],[ 2 2 2 1 3 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,-1,2,2,-1,-1,0,1,2,-1,1,1,2,1,0,1,2,3,-1] |
| Phi over symmetry | [-2,-2,1,1,1,1,-1,1,2,2,3,0,1,1,2,-1,-1,-1,-1,-1,0] |
| Phi of -K | [-2,-2,1,1,1,1,-1,1,2,2,3,0,1,1,2,-1,0,-1,1,-1,-1] |
| Phi of K* | [-1,-1,-1,-1,2,2,-1,-1,-1,2,3,-1,-1,1,2,0,0,1,1,2,-1] |
| Phi of -K* | [-2,-2,1,1,1,1,-1,1,2,2,3,0,1,1,2,-1,-1,-1,-1,-1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 2t^2-4t |
| Normalized Jones-Krushkal polynomial | 2z^2+19z+31 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2+19w^2z+31w |
| Inner characteristic polynomial | t^6+30t^4+13t^2+1 |
| Outer characteristic polynomial | t^7+42t^5+35t^3+5t |
| Flat arrow polynomial | -8*K1**2 + 4*K2 + 5 |
| 2-strand cable arrow polynomial | -192*K1**6 - 384*K1**4*K2**2 + 1312*K1**4*K2 - 3488*K1**4 + 320*K1**3*K2*K3 - 864*K1**3*K3 + 288*K1**2*K2**3 - 2720*K1**2*K2**2 - 192*K1**2*K2*K4 + 6656*K1**2*K2 - 192*K1**2*K3**2 - 2940*K1**2 - 160*K1*K2**2*K3 + 3088*K1*K2*K3 + 264*K1*K3*K4 - 160*K2**4 + 160*K2**2*K4 - 2464*K2**2 - 796*K3**2 - 88*K4**2 + 2550 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {2, 5}, {1, 3}], [{6}, {2, 5}, {4}, {1, 3}]] |
| If K is slice | False |