| Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,1,1,1,1,1,-1,0,1,1,0,0,1,1,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1669', '6.1775'] |
| Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.361', '6.460', '6.555', '6.651', '6.753', '6.782', '6.1029', '6.1197', '6.1200', '6.1232', '6.1236', '6.1278', '6.1281', '6.1343', '6.1380', '6.1385', '6.1389', '6.1484', '6.1492', '6.1493', '6.1527', '6.1533', '6.1550', '6.1553', '6.1557', '6.1576', '6.1578', '6.1582', '6.1586', '6.1674', '6.1698', '6.1754', '6.1759', '6.1775', '6.1791', '6.1798', '6.1800', '6.1805', '6.1822', '6.1826', '6.1839', '6.1844', '6.1845'] |
| Outer characteristic polynomial of the knot is: t^7+17t^5+25t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1669', '6.1775'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**6 - 448*K1**4*K2**2 + 1312*K1**4*K2 - 2944*K1**4 + 448*K1**3*K2*K3 - 736*K1**3*K3 - 192*K1**2*K2**4 + 1120*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 4704*K1**2*K2**2 - 544*K1**2*K2*K4 + 6464*K1**2*K2 - 448*K1**2*K3**2 - 48*K1**2*K4**2 - 2760*K1**2 + 224*K1*K2**3*K3 - 576*K1*K2**2*K3 - 32*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 4568*K1*K2*K3 + 608*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 680*K2**4 - 128*K2**2*K3**2 - 48*K2**2*K4**2 + 712*K2**2*K4 - 2366*K2**2 + 120*K2*K3*K5 + 16*K2*K4*K6 - 1096*K3**2 - 274*K4**2 - 32*K5**2 - 2*K6**2 + 2608 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1775'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4697', 'vk6.5002', 'vk6.6187', 'vk6.6660', 'vk6.8180', 'vk6.8600', 'vk6.9562', 'vk6.9903', 'vk6.17385', 'vk6.20918', 'vk6.20977', 'vk6.22328', 'vk6.22401', 'vk6.23550', 'vk6.23889', 'vk6.28398', 'vk6.36145', 'vk6.40056', 'vk6.40170', 'vk6.42107', 'vk6.43054', 'vk6.43360', 'vk6.46588', 'vk6.46679', 'vk6.48737', 'vk6.49537', 'vk6.49742', 'vk6.51435', 'vk6.55543', 'vk6.58924', 'vk6.65289', 'vk6.69774'] |
| 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 | O1O2O3U4U3O5O4U1U2O6U5U6 |
| R3 orbit | {'O1O2O3U4U3O5O4U1U2O6U5U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5O4U2U3O6O5U1U6 |
| Gauss code of K* | O1O2U3O4O3U1U2U5O6O5U4U6 |
| Gauss code of -K* | O1O2U1O3O4U5U2O6O5U6U3U4 |
| 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 1 0 0 1],[ 2 0 1 1 1 1 1],[ 0 -1 0 1 -1 0 1],[-1 -1 -1 0 -1 -1 0],[ 0 -1 1 1 0 0 0],[ 0 -1 0 1 0 0 1],[-1 -1 -1 0 0 -1 0]] |
| Primitive based matrix | [[ 0 1 1 0 0 0 -2],[-1 0 0 0 -1 -1 -1],[-1 0 0 -1 -1 -1 -1],[ 0 0 1 0 1 0 -1],[ 0 1 1 -1 0 0 -1],[ 0 1 1 0 0 0 -1],[ 2 1 1 1 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,0,0,0,2,0,0,1,1,1,1,1,1,1,-1,0,1,0,1,1] |
| Phi over symmetry | [-2,0,0,0,1,1,1,1,1,1,1,-1,0,1,1,0,0,1,1,1,0] |
| Phi of -K | [-2,0,0,0,1,1,1,1,1,2,2,-1,0,0,1,0,0,0,0,0,0] |
| Phi of K* | [-1,-1,0,0,0,2,0,0,0,0,2,0,0,1,2,0,-1,1,0,1,1] |
| Phi of -K* | [-2,0,0,0,1,1,1,1,1,1,1,-1,0,1,1,0,0,1,1,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 3z^2+20z+29 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+20w^2z+29w |
| Inner characteristic polynomial | t^6+11t^4+12t^2 |
| Outer characteristic polynomial | t^7+17t^5+25t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -256*K1**6 - 448*K1**4*K2**2 + 1312*K1**4*K2 - 2944*K1**4 + 448*K1**3*K2*K3 - 736*K1**3*K3 - 192*K1**2*K2**4 + 1120*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 4704*K1**2*K2**2 - 544*K1**2*K2*K4 + 6464*K1**2*K2 - 448*K1**2*K3**2 - 48*K1**2*K4**2 - 2760*K1**2 + 224*K1*K2**3*K3 - 576*K1*K2**2*K3 - 32*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 4568*K1*K2*K3 + 608*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 680*K2**4 - 128*K2**2*K3**2 - 48*K2**2*K4**2 + 712*K2**2*K4 - 2366*K2**2 + 120*K2*K3*K5 + 16*K2*K4*K6 - 1096*K3**2 - 274*K4**2 - 32*K5**2 - 2*K6**2 + 2608 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{6}, {3, 5}, {1, 4}, {2}]] |
| If K is slice | False |