| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,2,2,1,0,1,1,1,1,1,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1670'] |
| Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 8*K1*K2 + K1 + 6*K2 + 3*K3 + 7 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.906', '6.1223', '6.1338', '6.1351', '6.1571', '6.1670', '6.1718', '6.1743', '6.1765', '6.1793', '6.1852', '6.2070'] |
| Outer characteristic polynomial of the knot is: t^7+28t^5+30t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1670'] |
| 2-strand cable arrow polynomial of the knot is: -320*K1**4*K2**2 + 1088*K1**4*K2 - 2848*K1**4 + 640*K1**3*K2*K3 - 864*K1**3*K3 - 320*K1**2*K2**4 + 928*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 6752*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 256*K1**2*K2*K4 + 9728*K1**2*K2 - 928*K1**2*K3**2 - 6040*K1**2 + 736*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 1376*K1*K2**2*K3 - 416*K1*K2**2*K5 + 128*K1*K2*K3**3 - 288*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8488*K1*K2*K3 + 1080*K1*K3*K4 + 184*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1632*K2**4 - 32*K2**3*K6 - 992*K2**2*K3**2 - 64*K2**2*K4**2 + 1808*K2**2*K4 - 4530*K2**2 - 32*K2*K3**2*K4 + 792*K2*K3*K5 + 48*K2*K4*K6 - 32*K3**4 + 16*K3**2*K6 - 2332*K3**2 - 584*K4**2 - 172*K5**2 - 6*K6**2 + 4958 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1670'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16528', 'vk6.16621', 'vk6.17516', 'vk6.17572', 'vk6.18864', 'vk6.18940', 'vk6.19211', 'vk6.19504', 'vk6.23053', 'vk6.24118', 'vk6.25494', 'vk6.25567', 'vk6.26022', 'vk6.26406', 'vk6.34924', 'vk6.35038', 'vk6.36303', 'vk6.36373', 'vk6.37595', 'vk6.37682', 'vk6.42494', 'vk6.42607', 'vk6.43484', 'vk6.44611', 'vk6.54772', 'vk6.54864', 'vk6.56441', 'vk6.56557', 'vk6.59292', 'vk6.60178', 'vk6.66103', 'vk6.66143'] |
| 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 | O1O2O3U2O4U5U1U3O5O6U4U6 |
| R3 orbit | {'O1O2O3U2O4U5U1U3O5O6U4U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5O4O6U1U3U6O5U2 |
| Gauss code of K* | O1O2O3U1U4O5O4U2U6U3O6U5 |
| Gauss code of -K* | O1O2O3U4O5U1U5U2O6O4U6U3 |
| 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 2 1 -2 1],[ 1 0 0 2 1 0 1],[ 1 0 0 1 1 0 0],[-2 -2 -1 0 0 -2 1],[-1 -1 -1 0 0 -1 1],[ 2 0 0 2 1 0 1],[-1 -1 0 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 1 0 -1 -2 -2],[-1 -1 0 -1 0 -1 -1],[-1 0 1 0 -1 -1 -1],[ 1 1 0 1 0 0 0],[ 1 2 1 1 0 0 0],[ 2 2 1 1 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,-1,0,1,2,2,1,0,1,1,1,1,1,0,0,0] |
| Phi over symmetry | [-2,-1,-1,1,1,2,-1,0,1,2,2,1,0,1,1,1,1,1,0,0,0] |
| Phi of -K | [-2,-1,-1,1,1,2,1,1,2,2,2,0,1,1,1,1,2,2,-1,1,2] |
| Phi of K* | [-2,-1,-1,1,1,2,1,2,1,2,2,1,1,1,2,1,2,2,0,1,1] |
| Phi of -K* | [-2,-1,-1,1,1,2,0,0,1,1,2,0,0,1,1,1,1,2,-1,-1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 3z^2+23z+35 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+23w^2z+35w |
| Inner characteristic polynomial | t^6+16t^4+8t^2 |
| Outer characteristic polynomial | t^7+28t^5+30t^3+6t |
| Flat arrow polynomial | 4*K1**3 - 12*K1**2 - 8*K1*K2 + K1 + 6*K2 + 3*K3 + 7 |
| 2-strand cable arrow polynomial | -320*K1**4*K2**2 + 1088*K1**4*K2 - 2848*K1**4 + 640*K1**3*K2*K3 - 864*K1**3*K3 - 320*K1**2*K2**4 + 928*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 6752*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 256*K1**2*K2*K4 + 9728*K1**2*K2 - 928*K1**2*K3**2 - 6040*K1**2 + 736*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 1376*K1*K2**2*K3 - 416*K1*K2**2*K5 + 128*K1*K2*K3**3 - 288*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8488*K1*K2*K3 + 1080*K1*K3*K4 + 184*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1632*K2**4 - 32*K2**3*K6 - 992*K2**2*K3**2 - 64*K2**2*K4**2 + 1808*K2**2*K4 - 4530*K2**2 - 32*K2*K3**2*K4 + 792*K2*K3*K5 + 48*K2*K4*K6 - 32*K3**4 + 16*K3**2*K6 - 2332*K3**2 - 584*K4**2 - 172*K5**2 - 6*K6**2 + 4958 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}]] |
| If K is slice | True |