| Min(phi) over symmetries of the knot is: [-3,0,0,0,1,2,0,1,2,3,3,0,1,0,0,1,0,1,0,2,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.645'] |
| Arrow polynomial of the knot is: -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.217', '6.219', '6.304', '6.349', '6.390', '6.400', '6.416', '6.515', '6.518', '6.530', '6.582', '6.616', '6.629', '6.641', '6.645', '6.702', '6.710', '6.715', '6.729', '6.733', '6.734', '6.802', '6.840', '6.845', '6.854', '6.860', '6.900', '6.905', '6.921', '6.924', '6.979', '6.980', '6.996', '6.1044', '6.1067', '6.1086', '6.1100', '6.1139', '6.1145', '6.1149', '6.1167', '6.1169', '6.1183', '6.1314'] |
| Outer characteristic polynomial of the knot is: t^7+43t^5+84t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.645'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4 + 32*K1**3*K2*K3 - 1056*K1**2*K2**2 - 32*K1**2*K2*K4 + 1688*K1**2*K2 - 64*K1**2*K3**2 - 1640*K1**2 + 128*K1*K2**3*K3 - 192*K1*K2**2*K3 - 96*K1*K2**2*K5 + 1712*K1*K2*K3 + 192*K1*K3*K4 + 8*K1*K4*K5 - 120*K2**4 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 224*K2**2*K4 - 1158*K2**2 + 56*K2*K3*K5 + 8*K2*K4*K6 - 600*K3**2 - 106*K4**2 - 8*K5**2 - 2*K6**2 + 1160 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.645'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4225', 'vk6.4306', 'vk6.5490', 'vk6.5604', 'vk6.7589', 'vk6.7684', 'vk6.9090', 'vk6.9171', 'vk6.18378', 'vk6.18718', 'vk6.24831', 'vk6.25290', 'vk6.37019', 'vk6.37469', 'vk6.44188', 'vk6.44509', 'vk6.48545', 'vk6.48602', 'vk6.49246', 'vk6.49362', 'vk6.50332', 'vk6.50391', 'vk6.51071', 'vk6.51104', 'vk6.56159', 'vk6.56388', 'vk6.60688', 'vk6.61041', 'vk6.65827', 'vk6.66081', 'vk6.68816', 'vk6.69026', 'vk6.83732', 'vk6.83851', 'vk6.85070', 'vk6.85321', 'vk6.86662', 'vk6.86962', 'vk6.87430', 'vk6.89531'] |
| 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 | O1O2O3O4U1O5U4U5O6U3U2U6 |
| R3 orbit | {'O1O2O3O4U1O5U4U5O6U3U2U6', 'O1O2O3O4U5U3O6U4U2U1O5U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U3U2O5U6U1O6U4 |
| Gauss code of K* | O1O2O3U4U2U1U5O4U6O5O6U3 |
| Gauss code of -K* | O1O2O3U1O4O5U4O6U5U3U2U6 |
| 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 -3 0 0 0 1 2],[ 3 0 3 2 1 1 2],[ 0 -3 0 0 -1 1 2],[ 0 -2 0 0 -1 1 1],[ 0 -1 1 1 0 1 0],[-1 -1 -1 -1 -1 0 0],[-2 -2 -2 -1 0 0 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 0 -3],[-2 0 0 0 -1 -2 -2],[-1 0 0 -1 -1 -1 -1],[ 0 0 1 0 1 1 -1],[ 0 1 1 -1 0 0 -2],[ 0 2 1 -1 0 0 -3],[ 3 2 1 1 2 3 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,0,3,0,0,1,2,2,1,1,1,1,-1,-1,1,0,2,3] |
| Phi over symmetry | [-3,0,0,0,1,2,0,1,2,3,3,0,1,0,0,1,0,1,0,2,1] |
| Phi of -K | [-3,0,0,0,1,2,0,1,2,3,3,0,1,0,0,1,0,1,0,2,1] |
| Phi of K* | [-2,-1,0,0,0,3,1,0,1,2,3,0,0,0,3,0,-1,0,-1,1,2] |
| Phi of -K* | [-3,0,0,0,1,2,1,2,3,1,2,1,1,1,0,0,1,1,1,2,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 5z^2+18z+17 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+18w^2z+17w |
| Inner characteristic polynomial | t^6+29t^4+41t^2+1 |
| Outer characteristic polynomial | t^7+43t^5+84t^3+4t |
| Flat arrow polynomial | -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -64*K1**4 + 32*K1**3*K2*K3 - 1056*K1**2*K2**2 - 32*K1**2*K2*K4 + 1688*K1**2*K2 - 64*K1**2*K3**2 - 1640*K1**2 + 128*K1*K2**3*K3 - 192*K1*K2**2*K3 - 96*K1*K2**2*K5 + 1712*K1*K2*K3 + 192*K1*K3*K4 + 8*K1*K4*K5 - 120*K2**4 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 224*K2**2*K4 - 1158*K2**2 + 56*K2*K3*K5 + 8*K2*K4*K6 - 600*K3**2 - 106*K4**2 - 8*K5**2 - 2*K6**2 + 1160 |
| Genus of based matrix | 2 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{1, 6}, {5}, {4}, {3}, {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}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{5, 6}, {4}, {3}, {2}, {1}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {1, 5}, {4}, {3}, {2}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {2, 3}, {1}]] |
| If K is slice | False |