| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,1,2,0,0,1,0,1,0,1,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1517', '7.39384'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.568', '6.806', '6.1000', '6.1049', '6.1081', '6.1101', '6.1112', '6.1122', '6.1193', '6.1195', '6.1208', '6.1235', '6.1263', '6.1517', '6.1528', '6.1537', '6.1542', '6.1545', '6.1558', '6.1569', '6.1575', '6.1644', '6.1650', '6.1681', '6.1692', '6.1702', '6.1706', '6.1728', '6.1734', '6.1739', '6.1799', '6.1813', '6.1820', '6.1834', '6.1840', '6.1851', '6.1861', '6.1878'] |
| Outer characteristic polynomial of the knot is: t^7+20t^5+34t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1517', '7.39384'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**6 - 320*K1**4*K2**2 + 960*K1**4*K2 - 1936*K1**4 + 384*K1**3*K2*K3 - 192*K1**3*K3 - 192*K1**2*K2**4 + 736*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 3280*K1**2*K2**2 - 192*K1**2*K2*K4 + 3296*K1**2*K2 - 208*K1**2*K3**2 - 48*K1**2*K4**2 - 396*K1**2 + 224*K1*K2**3*K3 - 576*K1*K2**2*K3 - 64*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 2024*K1*K2*K3 + 248*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 408*K2**4 - 96*K2**2*K3**2 - 48*K2**2*K4**2 + 368*K2**2*K4 - 702*K2**2 + 72*K2*K3*K5 + 16*K2*K4*K6 - 284*K3**2 - 90*K4**2 - 16*K5**2 - 2*K6**2 + 832 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1517'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.311', 'vk6.350', 'vk6.422', 'vk6.700', 'vk6.747', 'vk6.828', 'vk6.869', 'vk6.1133', 'vk6.1500', 'vk6.1578', 'vk6.1675', 'vk6.1945', 'vk6.1984', 'vk6.2055', 'vk6.2168', 'vk6.2277', 'vk6.2650', 'vk6.2721', 'vk6.2789', 'vk6.3119', 'vk6.5244', 'vk6.6501', 'vk6.8873', 'vk6.9790', 'vk6.18310', 'vk6.18649', 'vk6.19406', 'vk6.19701', 'vk6.25200', 'vk6.25861', 'vk6.26188', 'vk6.28501', 'vk6.36929', 'vk6.37395', 'vk6.37972', 'vk6.39869', 'vk6.40285', 'vk6.44865', 'vk6.46941', 'vk6.49132'] |
| 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 | O1O2O3U2O4U3U1O5O6U5U6U4 |
| R3 orbit | {'O1O2O3U2O4U3U1O5O6U5U6U4', 'O1O2U1O3O4U2U3O5O6U5U6U4'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3U4U5U6O5O6U3U1O4U2 |
| Gauss code of K* | O1O2O3U4U5U6O5U3O6O4U1U2 |
| Gauss code of -K* | O1O2O3U2U3O4O5U1O6U5U6U4 |
| 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 0 2 -1 1],[ 1 0 -1 1 2 0 0],[ 1 1 0 1 1 0 0],[ 0 -1 -1 0 1 0 0],[-2 -2 -1 -1 0 -1 1],[ 1 0 0 0 1 0 1],[-1 0 0 0 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 1 -1 -1 -1 -2],[-1 -1 0 0 0 -1 0],[ 0 1 0 0 -1 0 -1],[ 1 1 0 1 0 0 1],[ 1 1 1 0 0 0 0],[ 1 2 0 1 -1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,-1,1,1,1,2,0,0,1,0,1,0,1,0,-1,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,1,1,1,2,0,0,1,0,1,0,1,0,-1,0] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,0,0,2,2,0,0,2,1,1,1,2,1,1,2] |
| Phi of K* | [-2,-1,0,1,1,1,2,1,1,2,2,1,2,1,2,0,1,0,0,-1,0] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,0,1,0,2,0,1,0,1,0,1,1,0,1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+16w^2z+21w |
| Inner characteristic polynomial | t^6+12t^4+15t^2 |
| Outer characteristic polynomial | t^7+20t^5+34t^3+3t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -256*K1**6 - 320*K1**4*K2**2 + 960*K1**4*K2 - 1936*K1**4 + 384*K1**3*K2*K3 - 192*K1**3*K3 - 192*K1**2*K2**4 + 736*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 3280*K1**2*K2**2 - 192*K1**2*K2*K4 + 3296*K1**2*K2 - 208*K1**2*K3**2 - 48*K1**2*K4**2 - 396*K1**2 + 224*K1*K2**3*K3 - 576*K1*K2**2*K3 - 64*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 2024*K1*K2*K3 + 248*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 408*K2**4 - 96*K2**2*K3**2 - 48*K2**2*K4**2 + 368*K2**2*K4 - 702*K2**2 + 72*K2*K3*K5 + 16*K2*K4*K6 - 284*K3**2 - 90*K4**2 - 16*K5**2 - 2*K6**2 + 832 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {5}, {2, 3}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {2, 3}, {1}], [{6}, {4, 5}, {2, 3}, {1}]] |
| If K is slice | False |