| Min(phi) over symmetries of the knot is: [-3,-1,-1,1,1,3,-1,1,1,3,4,1,0,1,1,0,1,2,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.675'] |
| Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.315', '6.337', '6.389', '6.418', '6.599', '6.675', '6.686', '6.688', '6.746', '6.747', '6.809', '6.1034', '6.1128', '6.1133', '6.1334', '6.1363', '6.1489', '6.1539', '6.1564', '6.1821', '6.1863'] |
| Outer characteristic polynomial of the knot is: t^7+59t^5+48t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.675'] |
| 2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 736*K1**4*K2 - 1344*K1**4 + 384*K1**3*K2*K3 - 224*K1**3*K3 - 320*K1**2*K2**4 + 1280*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 5920*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 256*K1**2*K2*K4 + 6160*K1**2*K2 - 192*K1**2*K3**2 - 3640*K1**2 + 1120*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 768*K1*K2**2*K3 - 192*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 4744*K1*K2*K3 + 368*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1456*K2**4 - 608*K2**2*K3**2 - 16*K2**2*K4**2 + 1080*K2**2*K4 - 2158*K2**2 + 312*K2*K3*K5 + 8*K2*K4*K6 - 1116*K3**2 - 280*K4**2 - 52*K5**2 - 2*K6**2 + 2822 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.675'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11595', 'vk6.11606', 'vk6.11946', 'vk6.11959', 'vk6.12941', 'vk6.12952', 'vk6.13252', 'vk6.20422', 'vk6.20432', 'vk6.21789', 'vk6.27778', 'vk6.27800', 'vk6.29300', 'vk6.31390', 'vk6.31409', 'vk6.32568', 'vk6.32587', 'vk6.32952', 'vk6.39202', 'vk6.39228', 'vk6.41426', 'vk6.47553', 'vk6.53194', 'vk6.53201', 'vk6.53507', 'vk6.57291', 'vk6.57293', 'vk6.61965', 'vk6.61971', 'vk6.64287', 'vk6.64290', 'vk6.64497'] |
| 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 | O1O2O3O4U3O5U1U2O6U5U6U4 |
| R3 orbit | {'O1O2O3O4U3O5U1U2O6U5U6U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U5U6O5U3U4O6U2 |
| Gauss code of K* | O1O2O3U4U5U6U3O6U1O4O5U2 |
| Gauss code of -K* | O1O2O3U2O4O5U3O6U1U6U4U5 |
| 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 -1 -1 3 1 1],[ 3 0 1 0 4 2 1],[ 1 -1 0 0 3 1 1],[ 1 0 0 0 1 0 0],[-3 -4 -3 -1 0 -1 1],[-1 -2 -1 0 1 0 1],[-1 -1 -1 0 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 1 1 -1 -1 -3],[-3 0 1 -1 -1 -3 -4],[-1 -1 0 -1 0 -1 -1],[-1 1 1 0 0 -1 -2],[ 1 1 0 0 0 0 0],[ 1 3 1 1 0 0 -1],[ 3 4 1 2 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,-1,1,1,3,-1,1,1,3,4,1,0,1,1,0,1,2,0,0,1] |
| Phi over symmetry | [-3,-1,-1,1,1,3,-1,1,1,3,4,1,0,1,1,0,1,2,0,0,1] |
| Phi of -K | [-3,-1,-1,1,1,3,1,2,2,3,2,0,1,1,1,2,2,3,-1,1,3] |
| Phi of K* | [-3,-1,-1,1,1,3,1,3,1,3,2,1,1,2,2,1,2,3,0,1,2] |
| Phi of -K* | [-3,-1,-1,1,1,3,0,1,1,2,4,0,0,0,1,1,1,3,-1,-1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 4z^2+21z+27 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+21w^2z+27w |
| Inner characteristic polynomial | t^6+37t^4+12t^2 |
| Outer characteristic polynomial | t^7+59t^5+48t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5 |
| 2-strand cable arrow polynomial | -384*K1**4*K2**2 + 736*K1**4*K2 - 1344*K1**4 + 384*K1**3*K2*K3 - 224*K1**3*K3 - 320*K1**2*K2**4 + 1280*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 5920*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 256*K1**2*K2*K4 + 6160*K1**2*K2 - 192*K1**2*K3**2 - 3640*K1**2 + 1120*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 768*K1*K2**2*K3 - 192*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 4744*K1*K2*K3 + 368*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1456*K2**4 - 608*K2**2*K3**2 - 16*K2**2*K4**2 + 1080*K2**2*K4 - 2158*K2**2 + 312*K2*K3*K5 + 8*K2*K4*K6 - 1116*K3**2 - 280*K4**2 - 52*K5**2 - 2*K6**2 + 2822 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{2, 6}, {3, 5}, {1, 4}]] |
| If K is slice | True |