| Min(phi) over symmetries of the knot is: [-3,0,0,0,1,2,1,1,2,0,3,-1,0,1,0,1,1,1,1,1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.589'] |
| Arrow polynomial of the knot is: 12*K1**3 - 6*K1**2 - 6*K1*K2 - 6*K1 + 3*K2 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.212', '6.358', '6.588', '6.589', '6.603', '6.990'] |
| Outer characteristic polynomial of the knot is: t^7+37t^5+73t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.589'] |
| 2-strand cable arrow polynomial of the knot is: -320*K1**4*K2**2 + 448*K1**4*K2 - 1696*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 352*K1**3*K2*K3 - 384*K1**3*K3 - 2112*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 4064*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 9632*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 480*K1**2*K2*K4 + 9208*K1**2*K2 - 160*K1**2*K3**2 - 5548*K1**2 + 2720*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1216*K1*K2**2*K3 - 128*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 7008*K1*K2*K3 + 296*K1*K3*K4 - 224*K2**6 + 160*K2**4*K4 - 2728*K2**4 - 1008*K2**2*K3**2 - 40*K2**2*K4**2 + 1224*K2**2*K4 - 2144*K2**2 + 200*K2*K3*K5 - 1516*K3**2 - 146*K4**2 + 3896 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.589'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3608', 'vk6.3673', 'vk6.3864', 'vk6.3991', 'vk6.7034', 'vk6.7069', 'vk6.7244', 'vk6.7365', 'vk6.17704', 'vk6.17753', 'vk6.24255', 'vk6.24316', 'vk6.36554', 'vk6.36631', 'vk6.43664', 'vk6.43771', 'vk6.48244', 'vk6.48313', 'vk6.48396', 'vk6.48423', 'vk6.50004', 'vk6.50039', 'vk6.50122', 'vk6.50147', 'vk6.55736', 'vk6.55793', 'vk6.60312', 'vk6.60395', 'vk6.65436', 'vk6.65465', 'vk6.68568', 'vk6.68597'] |
| 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 | O1O2O3O4U1O5U4O6U5U6U2U3 |
| R3 orbit | {'O1O2O3O4U1O5U4O6U5U6U2U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U2U3U5U6O5U1O6U4 |
| Gauss code of K* | O1O2O3O4U5U3U4U6O5U1O6U2 |
| Gauss code of -K* | O1O2O3O4U3O5U4O6U5U1U2U6 |
| 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 0 2 0 0 1],[ 3 0 2 3 1 1 0],[ 0 -2 0 1 -1 0 1],[-2 -3 -1 0 -1 0 1],[ 0 -1 1 1 0 1 1],[ 0 -1 0 0 -1 0 1],[-1 0 -1 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 0 -3],[-2 0 1 0 -1 -1 -3],[-1 -1 0 -1 -1 -1 0],[ 0 0 1 0 0 -1 -1],[ 0 1 1 0 0 -1 -2],[ 0 1 1 1 1 0 -1],[ 3 3 0 1 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,0,3,-1,0,1,1,3,1,1,1,0,0,1,1,1,2,1] |
| Phi over symmetry | [-3,0,0,0,1,2,1,1,2,0,3,-1,0,1,0,1,1,1,1,1,-1] |
| Phi of -K | [-3,0,0,0,1,2,1,2,2,4,2,0,1,0,1,1,0,2,0,1,2] |
| Phi of K* | [-2,-1,0,0,0,3,2,1,1,2,2,0,0,0,4,-1,0,1,1,2,2] |
| Phi of -K* | [-3,0,0,0,1,2,1,1,2,0,3,-1,0,1,0,1,1,1,1,1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 3z^2+20z+29 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2-4w^3z+24w^2z+29w |
| Inner characteristic polynomial | t^6+23t^4+24t^2 |
| Outer characteristic polynomial | t^7+37t^5+73t^3+8t |
| Flat arrow polynomial | 12*K1**3 - 6*K1**2 - 6*K1*K2 - 6*K1 + 3*K2 + 4 |
| 2-strand cable arrow polynomial | -320*K1**4*K2**2 + 448*K1**4*K2 - 1696*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 352*K1**3*K2*K3 - 384*K1**3*K3 - 2112*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 4064*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 9632*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 480*K1**2*K2*K4 + 9208*K1**2*K2 - 160*K1**2*K3**2 - 5548*K1**2 + 2720*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1216*K1*K2**2*K3 - 128*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 7008*K1*K2*K3 + 296*K1*K3*K4 - 224*K2**6 + 160*K2**4*K4 - 2728*K2**4 - 1008*K2**2*K3**2 - 40*K2**2*K4**2 + 1224*K2**2*K4 - 2144*K2**2 + 200*K2*K3*K5 - 1516*K3**2 - 146*K4**2 + 3896 |
| 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}], [{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}], [{3, 6}, {5}, {4}, {2}, {1}], [{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}], [{5, 6}, {4}, {3}, {2}, {1}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {1, 5}, {4}, {3}, {2}], [{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}, {3, 5}, {4}, {2}, {1}], [{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}, {4}, {1, 3}, {2}]] |
| If K is slice | False |