| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,0,2,3,3,-1,1,2,1,1,0,0,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.709'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['4.2', '6.303', '6.338', '6.381', '6.432', '6.468', '6.558', '6.583', '6.597', '6.607', '6.634', '6.637', '6.643', '6.654', '6.667', '6.701', '6.709', '6.712', '6.718', '6.728', '6.767', '6.801', '6.825', '6.827', '6.974', '6.994', '6.1042', '6.1061', '6.1069', '6.1181', '6.1271', '6.1286', '6.1287', '6.1289', '6.1297', '6.1337', '6.1355'] |
| Outer characteristic polynomial of the knot is: t^7+48t^5+116t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.709'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 352*K1**4*K2 - 384*K1**4 + 32*K1**3*K2*K3 - 32*K1**3*K3 - 256*K1**2*K2**4 + 992*K1**2*K2**3 + 32*K1**2*K2**2*K4 - 2752*K1**2*K2**2 - 128*K1**2*K2*K4 + 2688*K1**2*K2 - 1600*K1**2 + 416*K1*K2**3*K3 - 320*K1*K2**2*K3 - 64*K1*K2**2*K5 + 1472*K1*K2*K3 + 16*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 616*K2**4 - 128*K2**2*K3**2 - 8*K2**2*K4**2 + 288*K2**2*K4 - 624*K2**2 + 16*K2*K3*K5 - 160*K3**2 - 14*K4**2 + 972 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.709'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4191', 'vk6.4272', 'vk6.5441', 'vk6.5557', 'vk6.7554', 'vk6.7639', 'vk6.9060', 'vk6.9141', 'vk6.18235', 'vk6.18572', 'vk6.24707', 'vk6.25122', 'vk6.36830', 'vk6.37295', 'vk6.44070', 'vk6.44411', 'vk6.48511', 'vk6.48592', 'vk6.49207', 'vk6.49313', 'vk6.50302', 'vk6.50380', 'vk6.51065', 'vk6.51098', 'vk6.56038', 'vk6.56314', 'vk6.60591', 'vk6.60932', 'vk6.65704', 'vk6.66000', 'vk6.68749', 'vk6.68959', 'vk6.83496', 'vk6.83833', 'vk6.83968', 'vk6.85399', 'vk6.86321', 'vk6.87117', 'vk6.88318', 'vk6.88952'] |
| 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 | O1O2O3O4U5O6U4U1O5U2U3U6 |
| R3 orbit | {'O1O2O3O4U5O6U4U1O5U2U3U6', 'O1O2O3O4U5U3O5U6U1U2O6U4'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U2U3O6U4U1O5U6 |
| Gauss code of K* | O1O2O3U4U1U2U5O6U3O5O4U6 |
| Gauss code of -K* | O1O2O3U4O5O6U1O4U6U2U3U5 |
| 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 0 -1 3],[ 2 0 0 1 0 1 3],[ 1 0 0 1 1 0 2],[-1 -1 -1 0 1 -2 1],[ 0 0 -1 -1 0 0 0],[ 1 -1 0 2 0 0 3],[-3 -3 -2 -1 0 -3 0]] |
| Primitive based matrix | [[ 0 3 1 0 -1 -1 -2],[-3 0 -1 0 -2 -3 -3],[-1 1 0 1 -1 -2 -1],[ 0 0 -1 0 -1 0 0],[ 1 2 1 1 0 0 0],[ 1 3 2 0 0 0 -1],[ 2 3 1 0 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,1,1,2,1,0,2,3,3,-1,1,2,1,1,0,0,0,0,1] |
| Phi over symmetry | [-3,-1,0,1,1,2,1,0,2,3,3,-1,1,2,1,1,0,0,0,0,1] |
| Phi of -K | [-2,-1,-1,0,1,3,0,1,2,2,2,0,1,0,1,0,1,2,2,3,1] |
| Phi of K* | [-3,-1,0,1,1,2,1,3,1,2,2,2,0,1,2,1,0,2,0,0,1] |
| Phi of -K* | [-2,-1,-1,0,1,3,0,1,0,1,3,0,1,1,2,0,2,3,-1,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 4z^2+17z+19 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+17w^2z+19w |
| Inner characteristic polynomial | t^6+32t^4+67t^2+4 |
| Outer characteristic polynomial | t^7+48t^5+116t^3+7t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -192*K1**4*K2**2 + 352*K1**4*K2 - 384*K1**4 + 32*K1**3*K2*K3 - 32*K1**3*K3 - 256*K1**2*K2**4 + 992*K1**2*K2**3 + 32*K1**2*K2**2*K4 - 2752*K1**2*K2**2 - 128*K1**2*K2*K4 + 2688*K1**2*K2 - 1600*K1**2 + 416*K1*K2**3*K3 - 320*K1*K2**2*K3 - 64*K1*K2**2*K5 + 1472*K1*K2*K3 + 16*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 616*K2**4 - 128*K2**2*K3**2 - 8*K2**2*K4**2 + 288*K2**2*K4 - 624*K2**2 + 16*K2*K3*K5 - 160*K3**2 - 14*K4**2 + 972 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {5}, {3, 4}, {2}]] |
| If K is slice | False |