| Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,1,0,1,2,1,1,1,2,0,0,1,-1,-1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1112'] |
| 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+53t^5+51t^3+10t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1112'] |
| 2-strand cable arrow polynomial of the knot is: -512*K1**4*K2**2 + 960*K1**4*K2 - 1296*K1**4 - 512*K1**3*K2**2*K3 + 1312*K1**3*K2*K3 - 864*K1**3*K3 + 1728*K1**2*K2**3 - 6688*K1**2*K2**2 - 960*K1**2*K2*K4 + 5664*K1**2*K2 - 720*K1**2*K3**2 - 3716*K1**2 + 1472*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 960*K1*K2**2*K3 - 192*K1*K2**2*K5 - 32*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6512*K1*K2*K3 + 1080*K1*K3*K4 + 8*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 1368*K2**4 - 32*K2**3*K6 - 752*K2**2*K3**2 - 112*K2**2*K4**2 + 1096*K2**2*K4 - 2206*K2**2 + 216*K2*K3*K5 + 88*K2*K4*K6 - 1752*K3**2 - 470*K4**2 - 20*K5**2 - 18*K6**2 + 2956 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1112'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16338', 'vk6.16379', 'vk6.18066', 'vk6.18406', 'vk6.22673', 'vk6.22752', 'vk6.24509', 'vk6.24934', 'vk6.34615', 'vk6.34694', 'vk6.36646', 'vk6.37072', 'vk6.42308', 'vk6.42337', 'vk6.43928', 'vk6.44249', 'vk6.54609', 'vk6.54646', 'vk6.55886', 'vk6.56176', 'vk6.59098', 'vk6.59134', 'vk6.60406', 'vk6.60767', 'vk6.64640', 'vk6.64686', 'vk6.65516', 'vk6.65834', 'vk6.67995', 'vk6.68019', 'vk6.68602', 'vk6.68821'] |
| 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 | O1O2O3O4U2U5O6U3U4O5U1U6 |
| R3 orbit | {'O1O2O3O4U2U5O6U3U4O5U1U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U4O6U1U2O5U6U3 |
| Gauss code of K* | O1O2U3O4O5U2O6O3U6U1U4U5 |
| Gauss code of -K* | O1O2U3O4O5U1O3O6U4U5U6U2 |
| 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 -2 0 2 -1 2],[ 1 0 -2 1 3 -1 2],[ 2 2 0 1 2 0 2],[ 0 -1 -1 0 1 -1 1],[-2 -3 -2 -1 0 -2 0],[ 1 1 0 1 2 0 2],[-2 -2 -2 -1 0 -2 0]] |
| Primitive based matrix | [[ 0 2 2 0 -1 -1 -2],[-2 0 0 -1 -2 -2 -2],[-2 0 0 -1 -2 -3 -2],[ 0 1 1 0 -1 -1 -1],[ 1 2 2 1 0 1 0],[ 1 2 3 1 -1 0 -2],[ 2 2 2 1 0 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,0,1,1,2,0,1,2,2,2,1,2,3,2,1,1,1,-1,0,2] |
| Phi over symmetry | [-2,-2,0,1,1,2,0,1,0,1,2,1,1,1,2,0,0,1,-1,-1,1] |
| Phi of -K | [-2,-1,-1,0,2,2,-1,1,1,2,2,1,0,0,1,0,1,1,1,1,0] |
| Phi of K* | [-2,-2,0,1,1,2,0,1,0,1,2,1,1,1,2,0,0,1,-1,-1,1] |
| Phi of -K* | [-2,-1,-1,0,2,2,0,2,1,2,2,1,1,2,2,1,2,3,1,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 8z^2+29z+27 |
| Enhanced Jones-Krushkal polynomial | 8w^3z^2+29w^2z+27w |
| Inner characteristic polynomial | t^6+39t^4+24t^2 |
| Outer characteristic polynomial | t^7+53t^5+51t^3+10t |
| 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**2 + 960*K1**4*K2 - 1296*K1**4 - 512*K1**3*K2**2*K3 + 1312*K1**3*K2*K3 - 864*K1**3*K3 + 1728*K1**2*K2**3 - 6688*K1**2*K2**2 - 960*K1**2*K2*K4 + 5664*K1**2*K2 - 720*K1**2*K3**2 - 3716*K1**2 + 1472*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 960*K1*K2**2*K3 - 192*K1*K2**2*K5 - 32*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6512*K1*K2*K3 + 1080*K1*K3*K4 + 8*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 1368*K2**4 - 32*K2**3*K6 - 752*K2**2*K3**2 - 112*K2**2*K4**2 + 1096*K2**2*K4 - 2206*K2**2 + 216*K2*K3*K5 + 88*K2*K4*K6 - 1752*K3**2 - 470*K4**2 - 20*K5**2 - 18*K6**2 + 2956 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}]] |
| If K is slice | False |