| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,1,1,1,2,1,1,0,1,-1,1,1,0,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1572'] |
| 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+25t^5+41t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1572'] |
| 2-strand cable arrow polynomial of the knot is: -1024*K1**4*K2**2 + 1984*K1**4*K2 - 3136*K1**4 + 288*K1**3*K2*K3 - 320*K1**3*K3 - 320*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 2144*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 448*K1**2*K2**2*K4 - 7920*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 576*K1**2*K2*K4 + 8520*K1**2*K2 - 928*K1**2*K3**2 - 4572*K1**2 + 1280*K1*K2**3*K3 + 768*K1*K2**2*K3*K4 - 1472*K1*K2**2*K3 - 416*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 160*K1*K2*K3*K6 + 7064*K1*K2*K3 + 1296*K1*K3*K4 + 40*K1*K4*K5 - 32*K2**6 + 256*K2**4*K4 - 1456*K2**4 - 832*K2**2*K3**2 - 432*K2**2*K4**2 + 1384*K2**2*K4 - 3278*K2**2 + 504*K2*K3*K5 + 112*K2*K4*K6 + 24*K3**2*K6 - 1768*K3**2 - 520*K4**2 - 52*K5**2 - 18*K6**2 + 3982 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1572'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13375', 'vk6.13448', 'vk6.13637', 'vk6.13755', 'vk6.14157', 'vk6.14390', 'vk6.15619', 'vk6.16081', 'vk6.16461', 'vk6.16478', 'vk6.17635', 'vk6.22860', 'vk6.22893', 'vk6.24188', 'vk6.33130', 'vk6.33173', 'vk6.33235', 'vk6.33288', 'vk6.34837', 'vk6.34870', 'vk6.36434', 'vk6.42431', 'vk6.42448', 'vk6.43535', 'vk6.53559', 'vk6.53604', 'vk6.53635', 'vk6.53691', 'vk6.54719', 'vk6.55670', 'vk6.60223', 'vk6.64584'] |
| 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 | O1O2O3U4O5U1U5O6O4U6U2U3 |
| R3 orbit | {'O1O2O3U4O5U1U5O6O4U6U2U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U2U4O5O4U6U3O6U5 |
| Gauss code of K* | O1O2O3U4U2U3O5U6O4O6U1U5 |
| Gauss code of -K* | O1O2O3U4U3O5O6U5O4U1U2U6 |
| 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 2 0 1 -1],[ 2 0 1 2 1 1 -1],[ 0 -1 0 1 1 0 -1],[-2 -2 -1 0 -1 0 -1],[ 0 -1 -1 1 0 1 -1],[-1 -1 0 0 -1 0 0],[ 1 1 1 1 1 0 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 0 -1 -1 -1 -2],[-1 0 0 0 -1 0 -1],[ 0 1 0 0 1 -1 -1],[ 0 1 1 -1 0 -1 -1],[ 1 1 0 1 1 0 1],[ 2 2 1 1 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,0,1,1,1,2,0,1,0,1,-1,1,1,1,1,-1] |
| Phi over symmetry | [-2,-1,0,0,1,2,-1,1,1,1,2,1,1,0,1,-1,1,1,0,1,0] |
| Phi of -K | [-2,-1,0,0,1,2,2,1,1,2,2,0,0,2,2,-1,1,1,0,1,1] |
| Phi of K* | [-2,-1,0,0,1,2,1,1,1,2,2,0,1,2,2,-1,0,1,0,1,2] |
| Phi of -K* | [-2,-1,0,0,1,2,-1,1,1,1,2,1,1,0,1,-1,1,1,0,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 5z^2+26z+33 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+26w^2z+33w |
| Inner characteristic polynomial | t^6+15t^4+17t^2+1 |
| Outer characteristic polynomial | t^7+25t^5+41t^3+8t |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | -1024*K1**4*K2**2 + 1984*K1**4*K2 - 3136*K1**4 + 288*K1**3*K2*K3 - 320*K1**3*K3 - 320*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 2144*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 448*K1**2*K2**2*K4 - 7920*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 576*K1**2*K2*K4 + 8520*K1**2*K2 - 928*K1**2*K3**2 - 4572*K1**2 + 1280*K1*K2**3*K3 + 768*K1*K2**2*K3*K4 - 1472*K1*K2**2*K3 - 416*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 160*K1*K2*K3*K6 + 7064*K1*K2*K3 + 1296*K1*K3*K4 + 40*K1*K4*K5 - 32*K2**6 + 256*K2**4*K4 - 1456*K2**4 - 832*K2**2*K3**2 - 432*K2**2*K4**2 + 1384*K2**2*K4 - 3278*K2**2 + 504*K2*K3*K5 + 112*K2*K4*K6 + 24*K3**2*K6 - 1768*K3**2 - 520*K4**2 - 52*K5**2 - 18*K6**2 + 3982 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{5, 6}, {2, 4}, {1, 3}]] |
| If K is slice | False |