| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,1,3,1,2,1,1,0,1,1,0,1,1,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1305'] |
| 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+37t^5+95t^3+11t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1305'] |
| 2-strand cable arrow polynomial of the knot is: -896*K1**4*K2**2 + 1888*K1**4*K2 - 3312*K1**4 - 256*K1**3*K2**2*K3 + 480*K1**3*K2*K3 - 608*K1**3*K3 + 2304*K1**2*K2**3 - 7840*K1**2*K2**2 - 672*K1**2*K2*K4 + 8792*K1**2*K2 - 336*K1**2*K3**2 - 4796*K1**2 + 928*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 800*K1*K2**2*K3 - 224*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 6520*K1*K2*K3 + 816*K1*K3*K4 + 40*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 1584*K2**4 - 32*K2**3*K6 - 688*K2**2*K3**2 - 112*K2**2*K4**2 + 1056*K2**2*K4 - 3014*K2**2 - 96*K2*K3**2*K4 + 312*K2*K3*K5 + 88*K2*K4*K6 + 24*K3**2*K6 - 1592*K3**2 - 404*K4**2 - 52*K5**2 - 18*K6**2 + 3954 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1305'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13374', 'vk6.13447', 'vk6.13638', 'vk6.13756', 'vk6.14156', 'vk6.14389', 'vk6.15620', 'vk6.16082', 'vk6.16459', 'vk6.16476', 'vk6.17631', 'vk6.22858', 'vk6.22891', 'vk6.24186', 'vk6.33129', 'vk6.33172', 'vk6.33236', 'vk6.33289', 'vk6.34839', 'vk6.34872', 'vk6.36432', 'vk6.42433', 'vk6.42450', 'vk6.43533', 'vk6.53558', 'vk6.53603', 'vk6.53636', 'vk6.53692', 'vk6.54717', 'vk6.55672', 'vk6.60225', 'vk6.64583'] |
| 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 | O1O2O3U4O5O4U6U1O6U5U2U3 |
| R3 orbit | {'O1O2O3U4O5O4U6U1O6U5U2U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U2U4O5U3U5O6O4U6 |
| Gauss code of K* | O1O2O3U4U2U3O5U1U5O6O4U6 |
| Gauss code of -K* | O1O2O3U4O5O4U6U3O6U1U2U5 |
| 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 1 0 -1],[ 2 0 1 2 2 -1 2],[ 0 -1 0 1 1 -1 0],[-2 -2 -1 0 -1 -1 -2],[-1 -2 -1 1 0 0 -2],[ 0 1 1 1 0 0 0],[ 1 -2 0 2 2 0 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 -1 -1 -1 -2 -2],[-1 1 0 0 -1 -2 -2],[ 0 1 0 0 1 0 1],[ 0 1 1 -1 0 0 -1],[ 1 2 2 0 0 0 -2],[ 2 2 2 -1 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,1,1,1,2,2,0,1,2,2,-1,0,-1,0,1,2] |
| Phi over symmetry | [-2,-1,0,0,1,2,-1,1,3,1,2,1,1,0,1,1,0,1,1,1,0] |
| Phi of -K | [-2,-1,0,0,1,2,-1,1,3,1,2,1,1,0,1,1,0,1,1,1,0] |
| Phi of K* | [-2,-1,0,0,1,2,0,1,1,1,2,0,1,0,1,-1,1,1,1,3,-1] |
| Phi of -K* | [-2,-1,0,0,1,2,2,-1,1,2,2,0,0,2,2,1,0,1,1,1,1] |
| 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+27t^4+55t^2+4 |
| Outer characteristic polynomial | t^7+37t^5+95t^3+11t |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | -896*K1**4*K2**2 + 1888*K1**4*K2 - 3312*K1**4 - 256*K1**3*K2**2*K3 + 480*K1**3*K2*K3 - 608*K1**3*K3 + 2304*K1**2*K2**3 - 7840*K1**2*K2**2 - 672*K1**2*K2*K4 + 8792*K1**2*K2 - 336*K1**2*K3**2 - 4796*K1**2 + 928*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 800*K1*K2**2*K3 - 224*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 6520*K1*K2*K3 + 816*K1*K3*K4 + 40*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 1584*K2**4 - 32*K2**3*K6 - 688*K2**2*K3**2 - 112*K2**2*K4**2 + 1056*K2**2*K4 - 3014*K2**2 - 96*K2*K3**2*K4 + 312*K2*K3*K5 + 88*K2*K4*K6 + 24*K3**2*K6 - 1592*K3**2 - 404*K4**2 - 52*K5**2 - 18*K6**2 + 3954 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}]] |
| If K is slice | False |