| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,1,1,2,1,1,1,1,0,1,2,2,-1,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.790', '7.38019'] |
| Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.414', '6.594', '6.608', '6.790', '6.1233', '6.1285', '6.1293', '6.1513', '6.1752', '6.1787', '6.1810', '6.1818', '6.1867', '6.1868', '6.1923'] |
| Outer characteristic polynomial of the knot is: t^7+46t^5+38t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.790'] |
| 2-strand cable arrow polynomial of the knot is: -768*K1**6 - 1344*K1**4*K2**2 + 3040*K1**4*K2 - 3888*K1**4 + 1248*K1**3*K2*K3 - 608*K1**3*K3 - 768*K1**2*K2**4 + 2656*K1**2*K2**3 + 320*K1**2*K2**2*K4 - 7856*K1**2*K2**2 - 704*K1**2*K2*K4 + 6976*K1**2*K2 - 528*K1**2*K3**2 - 32*K1**2*K3*K5 - 48*K1**2*K4**2 - 1360*K1**2 + 1024*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1600*K1*K2**2*K3 - 256*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5024*K1*K2*K3 + 656*K1*K3*K4 + 120*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 128*K2**4*K4 - 1744*K2**4 - 32*K2**3*K6 - 560*K2**2*K3**2 - 96*K2**2*K4**2 + 1320*K2**2*K4 - 1348*K2**2 - 32*K2*K3**2*K4 + 320*K2*K3*K5 + 64*K2*K4*K6 + 8*K3**2*K6 - 768*K3**2 - 252*K4**2 - 56*K5**2 - 12*K6**2 + 2034 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.790'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70', 'vk6.127', 'vk6.220', 'vk6.269', 'vk6.294', 'vk6.676', 'vk6.1222', 'vk6.1271', 'vk6.1358', 'vk6.1407', 'vk6.1440', 'vk6.1920', 'vk6.2380', 'vk6.2440', 'vk6.2930', 'vk6.2982', 'vk6.5755', 'vk6.5788', 'vk6.7820', 'vk6.7853', 'vk6.13276', 'vk6.13307', 'vk6.14787', 'vk6.14794', 'vk6.15941', 'vk6.15950', 'vk6.18050', 'vk6.24490', 'vk6.33025', 'vk6.33389', 'vk6.43912', 'vk6.50501'] |
| 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 | O1O2O3O4U3O5U6U5U4O6U1U2 |
| R3 orbit | {'O1O2O3O4U3O5U6U5U4O6U1U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U3U4O5U1U6U5O6U2 |
| Gauss code of K* | O1O2O3U1O4O5U4U5U6U3O6U2 |
| Gauss code of -K* | O1O2O3U2O4U1U4U5U6O5O6U3 |
| 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 -1 1 -1 2 1 -2],[ 1 0 1 -1 2 1 -1],[-1 -1 0 -1 2 1 -3],[ 1 1 1 0 1 0 0],[-2 -2 -2 -1 0 0 -3],[-1 -1 -1 0 0 0 -1],[ 2 1 3 0 3 1 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 0 -2 -1 -2 -3],[-1 0 0 -1 0 -1 -1],[-1 2 1 0 -1 -1 -3],[ 1 1 0 1 0 1 0],[ 1 2 1 1 -1 0 -1],[ 2 3 1 3 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,0,2,1,2,3,1,0,1,1,1,1,3,-1,0,1] |
| Phi over symmetry | [-2,-1,-1,1,1,2,-1,1,1,2,1,1,1,1,0,1,2,2,-1,0,1] |
| Phi of -K | [-2,-1,-1,1,1,2,0,1,0,2,1,1,1,1,1,1,2,2,-1,-1,1] |
| Phi of K* | [-2,-1,-1,1,1,2,-1,1,1,2,1,1,1,1,0,1,2,2,-1,0,1] |
| Phi of -K* | [-2,-1,-1,1,1,2,0,1,1,3,3,1,0,1,1,1,1,2,-1,0,2] |
| 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+34t^4+18t^2+1 |
| Outer characteristic polynomial | t^7+46t^5+38t^3+5t |
| Flat arrow polynomial | 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5 |
| 2-strand cable arrow polynomial | -768*K1**6 - 1344*K1**4*K2**2 + 3040*K1**4*K2 - 3888*K1**4 + 1248*K1**3*K2*K3 - 608*K1**3*K3 - 768*K1**2*K2**4 + 2656*K1**2*K2**3 + 320*K1**2*K2**2*K4 - 7856*K1**2*K2**2 - 704*K1**2*K2*K4 + 6976*K1**2*K2 - 528*K1**2*K3**2 - 32*K1**2*K3*K5 - 48*K1**2*K4**2 - 1360*K1**2 + 1024*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1600*K1*K2**2*K3 - 256*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5024*K1*K2*K3 + 656*K1*K3*K4 + 120*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 128*K2**4*K4 - 1744*K2**4 - 32*K2**3*K6 - 560*K2**2*K3**2 - 96*K2**2*K4**2 + 1320*K2**2*K4 - 1348*K2**2 - 32*K2*K3**2*K4 + 320*K2*K3*K5 + 64*K2*K4*K6 + 8*K3**2*K6 - 768*K3**2 - 252*K4**2 - 56*K5**2 - 12*K6**2 + 2034 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{4, 6}, {3, 5}, {1, 2}]] |
| If K is slice | True |