| Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,0,1,2,3,3,1,2,1,2,-1,0,-1,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.698'] |
| 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+49t^5+99t^3+11t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.698'] |
| 2-strand cable arrow polynomial of the knot is: 1888*K1**4*K2 - 3200*K1**4 + 800*K1**3*K2*K3 - 1024*K1**3*K3 - 128*K1**2*K2**4 + 1120*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6656*K1**2*K2**2 - 800*K1**2*K2*K4 + 8024*K1**2*K2 - 768*K1**2*K3**2 - 32*K1**2*K4**2 - 4320*K1**2 + 352*K1*K2**3*K3 - 1536*K1*K2**2*K3 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 7040*K1*K2*K3 + 1296*K1*K3*K4 + 64*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1168*K2**4 - 288*K2**2*K3**2 - 8*K2**2*K4**2 + 1416*K2**2*K4 - 3424*K2**2 + 168*K2*K3*K5 - 1872*K3**2 - 548*K4**2 - 32*K5**2 + 3730 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.698'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10523', 'vk6.10530', 'vk6.10606', 'vk6.10621', 'vk6.10793', 'vk6.10810', 'vk6.10898', 'vk6.10907', 'vk6.19023', 'vk6.19034', 'vk6.19085', 'vk6.19097', 'vk6.19130', 'vk6.19142', 'vk6.25539', 'vk6.25552', 'vk6.25636', 'vk6.25647', 'vk6.25757', 'vk6.25769', 'vk6.30204', 'vk6.30211', 'vk6.30285', 'vk6.30300', 'vk6.30412', 'vk6.30429', 'vk6.37725', 'vk6.37744', 'vk6.56512', 'vk6.56515', 'vk6.66168', 'vk6.66179'] |
| 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 | O1O2O3O4U3O5U6U4O6U2U1U5 |
| R3 orbit | {'O1O2O3O4U3O5U6U4O6U2U1U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U4U3O6U1U6O5U2 |
| Gauss code of K* | O1O2O3U2U1U4U5O4U3O6O5U6 |
| Gauss code of -K* | O1O2O3U4O5O4U1O6U5U6U3U2 |
| 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 1 3 -1],[ 1 0 0 -1 2 3 0],[ 1 0 0 -1 2 2 0],[ 1 1 1 0 1 1 0],[-1 -2 -2 -1 0 0 -1],[-3 -3 -2 -1 0 0 -3],[ 1 0 0 0 1 3 0]] |
| Primitive based matrix | [[ 0 3 1 -1 -1 -1 -1],[-3 0 0 -1 -2 -3 -3],[-1 0 0 -1 -2 -1 -2],[ 1 1 1 0 1 0 1],[ 1 2 2 -1 0 0 0],[ 1 3 1 0 0 0 0],[ 1 3 2 -1 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,1,1,1,1,0,1,2,3,3,1,2,1,2,-1,0,-1,0,0,0] |
| Phi over symmetry | [-3,-1,1,1,1,1,0,1,2,3,3,1,2,1,2,-1,0,-1,0,0,0] |
| Phi of -K | [-1,-1,-1,-1,1,3,-1,-1,0,1,3,0,0,0,1,0,0,2,1,1,2] |
| Phi of K* | [-3,-1,1,1,1,1,2,1,1,2,3,0,1,0,1,0,0,-1,0,0,-1] |
| Phi of -K* | [-1,-1,-1,-1,1,3,-1,0,0,2,2,0,1,1,1,0,1,3,2,3,0] |
| 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+35t^4+55t^2+1 |
| Outer characteristic polynomial | t^7+49t^5+99t^3+11t |
| Flat arrow polynomial | 4*K1**3 - 2*K1*K2 - 2*K1 + 1 |
| 2-strand cable arrow polynomial | 1888*K1**4*K2 - 3200*K1**4 + 800*K1**3*K2*K3 - 1024*K1**3*K3 - 128*K1**2*K2**4 + 1120*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6656*K1**2*K2**2 - 800*K1**2*K2*K4 + 8024*K1**2*K2 - 768*K1**2*K3**2 - 32*K1**2*K4**2 - 4320*K1**2 + 352*K1*K2**3*K3 - 1536*K1*K2**2*K3 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 7040*K1*K2*K3 + 1296*K1*K3*K4 + 64*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1168*K2**4 - 288*K2**2*K3**2 - 8*K2**2*K4**2 + 1416*K2**2*K4 - 3424*K2**2 + 168*K2*K3*K5 - 1872*K3**2 - 548*K4**2 - 32*K5**2 + 3730 |
| 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}], [{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}, {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}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}]] |
| If K is slice | False |