| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,1,1,0,0,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1573'] |
| Arrow polynomial of the knot is: -2*K1**2 - 4*K1*K2 + 2*K1 + K2 + 2*K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.120', '6.213', '6.216', '6.320', '6.322', '6.615', '6.617', '6.891', '6.951', '6.955', '6.1001', '6.1012', '6.1022', '6.1043', '6.1047', '6.1063', '6.1074', '6.1249', '6.1544', '6.1546', '6.1555', '6.1573', '6.1574', '6.1585', '6.1756', '6.1757', '6.1762', '6.1802', '6.1803', '6.1824', '6.1881', '6.1935'] |
| Outer characteristic polynomial of the knot is: t^7+20t^5+28t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1573'] |
| 2-strand cable arrow polynomial of the knot is: -2544*K1**4 + 352*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1056*K1**3*K3 + 96*K1**2*K2**2*K4 - 2528*K1**2*K2**2 - 640*K1**2*K2*K4 + 6624*K1**2*K2 - 1168*K1**2*K3**2 - 224*K1**2*K3*K5 - 64*K1**2*K4**2 - 4116*K1**2 - 736*K1*K2**2*K3 - 160*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 5984*K1*K2*K3 + 2056*K1*K3*K4 + 432*K1*K4*K5 - 360*K2**4 - 112*K2**2*K3**2 - 16*K2**2*K4**2 + 1184*K2**2*K4 - 3588*K2**2 + 432*K2*K3*K5 + 32*K2*K4*K6 - 2160*K3**2 - 950*K4**2 - 244*K5**2 - 12*K6**2 + 3708 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1573'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13391', 'vk6.13480', 'vk6.13669', 'vk6.13771', 'vk6.14205', 'vk6.14454', 'vk6.15679', 'vk6.16129', 'vk6.16757', 'vk6.16771', 'vk6.16892', 'vk6.19045', 'vk6.19303', 'vk6.19596', 'vk6.23170', 'vk6.23273', 'vk6.25658', 'vk6.26491', 'vk6.33146', 'vk6.33205', 'vk6.33300', 'vk6.35157', 'vk6.35190', 'vk6.37750', 'vk6.42660', 'vk6.42677', 'vk6.42793', 'vk6.44719', 'vk6.53575', 'vk6.53703', 'vk6.54968', 'vk6.64619'] |
| 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 | O1O2O3U4O5U1U5O6O4U6U3U2 |
| R3 orbit | {'O1O2O3U4O5U1U5O6O4U6U3U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U2U1U4O5O4U6U3O6U5 |
| Gauss code of K* | O1O2O3U4U3U2O5U6O4O6U1U5 |
| Gauss code of -K* | O1O2O3U4U3O5O6U5O4U2U1U6 |
| 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 1 1 0 1 -1],[ 2 0 2 1 1 1 -1],[-1 -2 0 0 0 0 -1],[-1 -1 0 0 0 0 -1],[ 0 -1 0 0 0 1 -1],[-1 -1 0 0 -1 0 0],[ 1 1 1 1 1 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 0 0 0 -1 -1],[-1 0 0 0 0 -1 -2],[-1 0 0 0 -1 0 -1],[ 0 0 0 1 0 -1 -1],[ 1 1 1 0 1 0 1],[ 2 1 2 1 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,0,0,0,1,1,0,0,1,2,1,0,1,1,1,-1] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,1,1,0,0,0,0,0] |
| Phi of -K | [-2,-1,0,1,1,1,2,1,1,2,2,0,1,1,2,1,1,0,0,0,0] |
| Phi of K* | [-1,-1,-1,0,1,2,0,0,0,2,2,0,1,1,1,1,1,2,0,1,2] |
| Phi of -K* | [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,1,1,0,0,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 8z^2+27z+23 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+12w^3z^2+27w^2z+23w |
| Inner characteristic polynomial | t^6+12t^4+11t^2+1 |
| Outer characteristic polynomial | t^7+20t^5+28t^3+6t |
| Flat arrow polynomial | -2*K1**2 - 4*K1*K2 + 2*K1 + K2 + 2*K3 + 2 |
| 2-strand cable arrow polynomial | -2544*K1**4 + 352*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1056*K1**3*K3 + 96*K1**2*K2**2*K4 - 2528*K1**2*K2**2 - 640*K1**2*K2*K4 + 6624*K1**2*K2 - 1168*K1**2*K3**2 - 224*K1**2*K3*K5 - 64*K1**2*K4**2 - 4116*K1**2 - 736*K1*K2**2*K3 - 160*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 5984*K1*K2*K3 + 2056*K1*K3*K4 + 432*K1*K4*K5 - 360*K2**4 - 112*K2**2*K3**2 - 16*K2**2*K4**2 + 1184*K2**2*K4 - 3588*K2**2 + 432*K2*K3*K5 + 32*K2*K4*K6 - 2160*K3**2 - 950*K4**2 - 244*K5**2 - 12*K6**2 + 3708 |
| 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}], [{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}], [{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}, {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}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]] |
| If K is slice | False |