| Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,-1,1,4,2,4,1,2,0,1,2,1,3,2,2,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.744'] |
| Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.209', '6.231', '6.391', '6.419', '6.600', '6.661', '6.744', '6.812', '6.826', '6.1114', '6.1125', '6.1202', '6.1275', '6.1292', '6.1305', '6.1322', '6.1365', '6.1481', '6.1483', '6.1497', '6.1543', '6.1549', '6.1572', '6.1577', '6.1580', '6.1594', '6.1641', '6.1658', '6.1683', '6.1753', '6.1830', '6.1907', '6.1928'] |
| Outer characteristic polynomial of the knot is: t^7+94t^5+114t^3+14t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.744'] |
| 2-strand cable arrow polynomial of the knot is: -448*K1**4*K2**2 + 640*K1**4*K2 - 976*K1**4 + 128*K1**3*K2**3*K3 - 384*K1**3*K2**2*K3 + 1056*K1**3*K2*K3 - 896*K1**3*K3 - 320*K1**2*K2**4 + 3136*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 10176*K1**2*K2**2 - 640*K1**2*K2*K4 + 8784*K1**2*K2 - 400*K1**2*K3**2 - 5640*K1**2 + 1792*K1*K2**3*K3 - 1408*K1*K2**2*K3 - 224*K1*K2**2*K5 + 32*K1*K2*K3**3 - 160*K1*K2*K3*K4 + 8392*K1*K2*K3 + 336*K1*K3*K4 + 56*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 2864*K2**4 - 1024*K2**2*K3**2 - 16*K2**2*K4**2 + 1664*K2**2*K4 - 2502*K2**2 + 408*K2*K3*K5 + 8*K2*K4*K6 - 1876*K3**2 - 216*K4**2 - 60*K5**2 - 2*K6**2 + 3926 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.744'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4725', 'vk6.5044', 'vk6.6254', 'vk6.6702', 'vk6.8222', 'vk6.8662', 'vk6.9606', 'vk6.9931', 'vk6.20295', 'vk6.21628', 'vk6.27587', 'vk6.29139', 'vk6.39009', 'vk6.41257', 'vk6.45773', 'vk6.47450', 'vk6.48757', 'vk6.48958', 'vk6.49558', 'vk6.49772', 'vk6.50767', 'vk6.50971', 'vk6.51246', 'vk6.51453', 'vk6.57154', 'vk6.58338', 'vk6.61776', 'vk6.62895', 'vk6.66775', 'vk6.67651', 'vk6.69419', 'vk6.70141'] |
| 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 | O1O2O3O4U2O5U1U6U5O6U3U4 |
| R3 orbit | {'O1O2O3O4U2O5U1U6U5O6U3U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U2O5U6U5U4O6U3 |
| Gauss code of K* | O1O2O3U2O4O5U1U6U4U5O6U3 |
| Gauss code of -K* | O1O2O3U1O4U5U6U4U3O5O6U2 |
| 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 -2 1 3 2 -1],[ 3 0 0 3 4 1 2],[ 2 0 0 1 2 0 2],[-1 -3 -1 0 1 1 -2],[-3 -4 -2 -1 0 1 -4],[-2 -1 0 -1 -1 0 -2],[ 1 -2 -2 2 4 2 0]] |
| Primitive based matrix | [[ 0 3 2 1 -1 -2 -3],[-3 0 1 -1 -4 -2 -4],[-2 -1 0 -1 -2 0 -1],[-1 1 1 0 -2 -1 -3],[ 1 4 2 2 0 -2 -2],[ 2 2 0 1 2 0 0],[ 3 4 1 3 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-1,1,2,3,-1,1,4,2,4,1,2,0,1,2,1,3,2,2,0] |
| Phi over symmetry | [-3,-2,-1,1,2,3,-1,1,4,2,4,1,2,0,1,2,1,3,2,2,0] |
| Phi of -K | [-3,-2,-1,1,2,3,1,0,1,4,2,-1,2,4,3,0,1,0,0,1,2] |
| Phi of K* | [-3,-2,-1,1,2,3,2,1,0,3,2,0,1,4,4,0,2,1,-1,0,1] |
| Phi of -K* | [-3,-2,-1,1,2,3,0,2,3,1,4,2,1,0,2,2,2,4,1,1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 8z^2+29z+27 |
| Enhanced Jones-Krushkal polynomial | 8w^3z^2+29w^2z+27w |
| Inner characteristic polynomial | t^6+66t^4+66t^2+4 |
| Outer characteristic polynomial | t^7+94t^5+114t^3+14t |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | -448*K1**4*K2**2 + 640*K1**4*K2 - 976*K1**4 + 128*K1**3*K2**3*K3 - 384*K1**3*K2**2*K3 + 1056*K1**3*K2*K3 - 896*K1**3*K3 - 320*K1**2*K2**4 + 3136*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 10176*K1**2*K2**2 - 640*K1**2*K2*K4 + 8784*K1**2*K2 - 400*K1**2*K3**2 - 5640*K1**2 + 1792*K1*K2**3*K3 - 1408*K1*K2**2*K3 - 224*K1*K2**2*K5 + 32*K1*K2*K3**3 - 160*K1*K2*K3*K4 + 8392*K1*K2*K3 + 336*K1*K3*K4 + 56*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 2864*K2**4 - 1024*K2**2*K3**2 - 16*K2**2*K4**2 + 1664*K2**2*K4 - 2502*K2**2 + 408*K2*K3*K5 + 8*K2*K4*K6 - 1876*K3**2 - 216*K4**2 - 60*K5**2 - 2*K6**2 + 3926 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{3, 6}, {2, 5}, {1, 4}]] |
| If K is slice | True |