| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,0,1,2,1,1,1,0,0,0,1,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1799'] |
| 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+51t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1799'] |
| 2-strand cable arrow polynomial of the knot is: 512*K1**4*K2**3 - 768*K1**4*K2**2 + 480*K1**4*K2 - 688*K1**4 - 512*K1**3*K2**2*K3 + 896*K1**3*K2*K3 - 832*K1**3*K3 - 1280*K1**2*K2**4 + 2848*K1**2*K2**3 - 5248*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 640*K1**2*K2*K4 + 4808*K1**2*K2 - 304*K1**2*K3**2 - 3452*K1**2 + 1440*K1*K2**3*K3 - 480*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 4240*K1*K2*K3 + 376*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 1336*K2**4 - 496*K2**2*K3**2 - 48*K2**2*K4**2 + 432*K2**2*K4 - 1198*K2**2 + 88*K2*K3*K5 + 16*K2*K4*K6 - 1048*K3**2 - 142*K4**2 - 4*K5**2 - 2*K6**2 + 2244 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1799'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19911', 'vk6.19964', 'vk6.21132', 'vk6.21224', 'vk6.26828', 'vk6.26965', 'vk6.28608', 'vk6.28703', 'vk6.38260', 'vk6.38377', 'vk6.40382', 'vk6.40541', 'vk6.45127', 'vk6.45249', 'vk6.46985', 'vk6.47054', 'vk6.56687', 'vk6.56766', 'vk6.57764', 'vk6.57883', 'vk6.61082', 'vk6.61240', 'vk6.62345', 'vk6.62449', 'vk6.66380', 'vk6.66472', 'vk6.67138', 'vk6.67257', 'vk6.69037', 'vk6.69122', 'vk6.69829', 'vk6.69890'] |
| 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 | O1O2O3U4U3O5O6U1U2O4U6U5 |
| R3 orbit | {'O1O2O3U4U3O5O6U1U2O4U6U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5O6U2U3O5O4U1U6 |
| Gauss code of K* | O1O2U3O4O5U1U2U6O3O6U5U4 |
| Gauss code of -K* | O1O2U3O4O5U2U1O6O3U6U4U5 |
| 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 0 1 -1 1 1],[ 2 0 1 0 0 2 1],[ 0 -1 0 0 -1 1 0],[-1 0 0 0 -1 -1 -1],[ 1 0 1 1 0 0 1],[-1 -2 -1 1 0 0 0],[-1 -1 0 1 -1 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 0 -1 -1],[-1 -1 0 -1 0 -1 0],[-1 0 1 0 -1 0 -2],[ 0 0 0 1 0 -1 -1],[ 1 1 1 0 1 0 0],[ 2 1 0 2 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,0,1,1,1,0,1,0,1,0,2,1,1,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,1,0,1,2,1,1,1,0,0,0,1,-1,-1,0] |
| Phi of -K | [-2,-1,0,1,1,1,1,1,1,2,3,0,2,1,1,0,1,1,0,-1,-1] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,-1,1,1,3,0,0,2,1,1,1,2,0,1,1] |
| Phi of -K* | [-2,-1,0,1,1,1,0,1,0,1,2,1,1,1,0,0,0,1,-1,-1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 4z^2+17z+19 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+6w^3z^2-6w^3z+23w^2z+19w |
| Inner characteristic polynomial | t^6+12t^4+26t^2+1 |
| Outer characteristic polynomial | t^7+20t^5+51t^3+8t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | 512*K1**4*K2**3 - 768*K1**4*K2**2 + 480*K1**4*K2 - 688*K1**4 - 512*K1**3*K2**2*K3 + 896*K1**3*K2*K3 - 832*K1**3*K3 - 1280*K1**2*K2**4 + 2848*K1**2*K2**3 - 5248*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 640*K1**2*K2*K4 + 4808*K1**2*K2 - 304*K1**2*K3**2 - 3452*K1**2 + 1440*K1*K2**3*K3 - 480*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 4240*K1*K2*K3 + 376*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 1336*K2**4 - 496*K2**2*K3**2 - 48*K2**2*K4**2 + 432*K2**2*K4 - 1198*K2**2 + 88*K2*K3*K5 + 16*K2*K4*K6 - 1048*K3**2 - 142*K4**2 - 4*K5**2 - 2*K6**2 + 2244 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {3, 4}, {1, 2}]] |
| If K is slice | False |