| Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,0,1,1,2,1,1,2,2,0,1,2,0,1,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.1251'] |
| Arrow polynomial of the knot is: -8*K1**4 + 8*K1**2*K2 + 4*K1**2 - 2*K1*K3 - K2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.133', '6.517', '6.545', '6.1198', '6.1251', '6.1906'] |
| Outer characteristic polynomial of the knot is: t^7+41t^5+91t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1251'] |
| 2-strand cable arrow polynomial of the knot is: 2144*K1**2*K2**3 - 3968*K1**2*K2**2 - 256*K1**2*K2*K4 + 3192*K1**2*K2 - 2560*K1**2 - 1152*K1*K2**4*K3 + 1696*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 1344*K1*K2**2*K3 - 64*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 3600*K1*K2*K3 + 392*K1*K3*K4 + 24*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1088*K2**6 - 128*K2**5*K6 - 64*K2**4*K4**2 + 1824*K2**4*K4 - 3360*K2**4 + 64*K2**3*K4*K6 - 128*K2**3*K6 - 1376*K2**2*K3**2 - 544*K2**2*K4**2 + 2136*K2**2*K4 - 8*K2**2*K6**2 - 552*K2**2 + 568*K2*K3*K5 + 112*K2*K4*K6 - 1000*K3**2 - 442*K4**2 - 88*K5**2 - 8*K6**2 + 2056 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1251'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73191', 'vk6.73205', 'vk6.73629', 'vk6.74309', 'vk6.74403', 'vk6.74953', 'vk6.75012', 'vk6.75105', 'vk6.75122', 'vk6.75567', 'vk6.75591', 'vk6.76521', 'vk6.76585', 'vk6.76932', 'vk6.78047', 'vk6.78065', 'vk6.78535', 'vk6.78562', 'vk6.79357', 'vk6.79779', 'vk6.79848', 'vk6.79965', 'vk6.80815', 'vk6.80880', 'vk6.83694', 'vk6.84700', 'vk6.84811', 'vk6.85274', 'vk6.85635', 'vk6.87708', 'vk6.88378', 'vk6.89484'] |
| 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 | O1O2O3O4U5U1U2O5O6U4U3U6 |
| R3 orbit | {'O1O2O3O4U5U1U2O5O6U4U3U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U2U1O5O6U3U4U6 |
| Gauss code of K* | O1O2O3U1U4O5O6O4U2U3U6U5 |
| Gauss code of -K* | O1O2O3U1U4O5O6O4U3U2U5U6 |
| 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 1 -2 2],[ 2 0 1 2 1 1 2],[ 0 -1 0 1 0 0 2],[-1 -2 -1 0 0 -1 2],[-1 -1 0 0 0 -1 1],[ 2 -1 0 1 1 0 2],[-2 -2 -2 -2 -1 -2 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -2 -2],[-2 0 -1 -2 -2 -2 -2],[-1 1 0 0 0 -1 -1],[-1 2 0 0 -1 -1 -2],[ 0 2 0 1 0 0 -1],[ 2 2 1 1 0 0 -1],[ 2 2 1 2 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,2,2,1,2,2,2,2,0,0,1,1,1,1,2,0,1,1] |
| Phi over symmetry | [-2,-2,0,1,1,2,-1,0,1,1,2,1,1,2,2,0,1,2,0,1,2] |
| Phi of -K | [-2,-2,0,1,1,2,-1,1,1,2,2,2,2,2,2,0,1,0,0,-1,0] |
| Phi of K* | [-2,-1,-1,0,2,2,-1,0,0,2,2,0,0,1,2,1,2,2,1,2,1] |
| Phi of -K* | [-2,-2,0,1,1,2,-1,0,1,1,2,1,1,2,2,0,1,2,0,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 6z^2+19z+15 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+10w^3z^2-4w^3z+23w^2z+15w |
| Inner characteristic polynomial | t^6+27t^4+12t^2 |
| Outer characteristic polynomial | t^7+41t^5+91t^3+7t |
| Flat arrow polynomial | -8*K1**4 + 8*K1**2*K2 + 4*K1**2 - 2*K1*K3 - K2 |
| 2-strand cable arrow polynomial | 2144*K1**2*K2**3 - 3968*K1**2*K2**2 - 256*K1**2*K2*K4 + 3192*K1**2*K2 - 2560*K1**2 - 1152*K1*K2**4*K3 + 1696*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 1344*K1*K2**2*K3 - 64*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 3600*K1*K2*K3 + 392*K1*K3*K4 + 24*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1088*K2**6 - 128*K2**5*K6 - 64*K2**4*K4**2 + 1824*K2**4*K4 - 3360*K2**4 + 64*K2**3*K4*K6 - 128*K2**3*K6 - 1376*K2**2*K3**2 - 544*K2**2*K4**2 + 2136*K2**2*K4 - 8*K2**2*K6**2 - 552*K2**2 + 568*K2*K3*K5 + 112*K2*K4*K6 - 1000*K3**2 - 442*K4**2 - 88*K5**2 - 8*K6**2 + 2056 |
| 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}], [{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}], [{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}], [{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}, {2, 5}, {4}, {3}, {1}], [{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}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}]] |
| If K is slice | False |