| Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,0,1,2,2,3,1,1,1,1,-1,-1,-1,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.382'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1*K2 - 2*K1 + 1 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.233', '6.340', '6.382', '6.550', '6.656', '6.663', '6.683', '6.698', '6.739', '6.745', '6.759', '6.765', '6.1357', '6.1358', '6.1370'] |
| Outer characteristic polynomial of the knot is: t^7+39t^5+30t^3+2t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.382'] |
| 2-strand cable arrow polynomial of the knot is: 2528*K1**4*K2 - 5824*K1**4 + 1440*K1**3*K2*K3 - 1280*K1**3*K3 - 128*K1**2*K2**4 + 480*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5632*K1**2*K2**2 - 544*K1**2*K2*K4 + 7264*K1**2*K2 - 1408*K1**2*K3**2 - 32*K1**2*K4**2 - 1056*K1**2 + 224*K1*K2**3*K3 - 416*K1*K2**2*K3 - 64*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 4456*K1*K2*K3 + 792*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 432*K2**4 - 128*K2**2*K3**2 - 8*K2**2*K4**2 + 336*K2**2*K4 - 1920*K2**2 + 72*K2*K3*K5 - 792*K3**2 - 100*K4**2 + 2122 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.382'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10521', 'vk6.10525', 'vk6.10602', 'vk6.10610', 'vk6.10789', 'vk6.10797', 'vk6.10896', 'vk6.10900', 'vk6.19022', 'vk6.19038', 'vk6.19096', 'vk6.19098', 'vk6.19143', 'vk6.19145', 'vk6.25540', 'vk6.25556', 'vk6.25635', 'vk6.25651', 'vk6.25768', 'vk6.25770', 'vk6.30206', 'vk6.30210', 'vk6.30289', 'vk6.30297', 'vk6.30416', 'vk6.30424', 'vk6.37724', 'vk6.37740', 'vk6.56511', 'vk6.56519', 'vk6.66167', 'vk6.66175'] |
| 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 | O1O2O3O4O5U4U1O6U5U6U3U2 |
| R3 orbit | {'O1O2O3O4O5U4U1O6U5U6U3U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U4U3U6U1O6U5U2 |
| Gauss code of K* | O1O2O3O4U5U4U3U6U1O6O5U2 |
| Gauss code of -K* | O1O2O3O4U3O5O6U4U6U2U1U5 |
| 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 -3 1 1 -1 1 1],[ 3 0 3 2 0 2 1],[-1 -3 0 0 -1 0 1],[-1 -2 0 0 -1 0 1],[ 1 0 1 1 0 1 1],[-1 -2 0 0 -1 0 1],[-1 -1 -1 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 1 -1 -3],[-1 0 1 0 0 -1 -2],[-1 -1 0 -1 -1 -1 -1],[-1 0 1 0 0 -1 -2],[-1 0 1 0 0 -1 -3],[ 1 1 1 1 1 0 0],[ 3 2 1 2 3 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,-1,1,3,-1,0,0,1,2,1,1,1,1,0,1,2,1,3,0] |
| Phi over symmetry | [-3,-1,1,1,1,1,0,1,2,2,3,1,1,1,1,-1,-1,-1,0,0,0] |
| Phi of -K | [-3,-1,1,1,1,1,2,1,2,2,3,1,1,1,1,0,0,-1,0,-1,-1] |
| Phi of K* | [-1,-1,-1,-1,1,3,-1,-1,-1,1,3,0,0,1,1,0,1,2,1,2,2] |
| Phi of -K* | [-3,-1,1,1,1,1,0,1,2,2,3,1,1,1,1,-1,-1,-1,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-3t |
| Normalized Jones-Krushkal polynomial | 8z^2+29z+27 |
| Enhanced Jones-Krushkal polynomial | 8w^3z^2+29w^2z+27w |
| Inner characteristic polynomial | t^6+25t^4+10t^2 |
| Outer characteristic polynomial | t^7+39t^5+30t^3+2t |
| Flat arrow polynomial | 4*K1**3 - 2*K1*K2 - 2*K1 + 1 |
| 2-strand cable arrow polynomial | 2528*K1**4*K2 - 5824*K1**4 + 1440*K1**3*K2*K3 - 1280*K1**3*K3 - 128*K1**2*K2**4 + 480*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5632*K1**2*K2**2 - 544*K1**2*K2*K4 + 7264*K1**2*K2 - 1408*K1**2*K3**2 - 32*K1**2*K4**2 - 1056*K1**2 + 224*K1*K2**3*K3 - 416*K1*K2**2*K3 - 64*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 4456*K1*K2*K3 + 792*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 432*K2**4 - 128*K2**2*K3**2 - 8*K2**2*K4**2 + 336*K2**2*K4 - 1920*K2**2 + 72*K2*K3*K5 - 792*K3**2 - 100*K4**2 + 2122 |
| Genus of based matrix | 2 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{1, 6}, {5}, {4}, {3}, {2}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{2, 6}, {5}, {4}, {3}, {1}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{4, 6}, {5}, {3}, {2}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {3}, {1, 2}]] |
| If K is slice | False |