| Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,-1,1,2,3,3,1,2,2,2,0,0,1,0,2,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.120'] |
| 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+70t^5+51t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.120'] |
| 2-strand cable arrow polynomial of the knot is: -1232*K1**2*K2**2 - 160*K1**2*K2*K4 + 1664*K1**2*K2 - 16*K1**2*K3**2 - 1716*K1**2 + 96*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 384*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 2224*K1*K2*K3 + 456*K1*K3*K4 + 8*K1*K5*K6 - 72*K2**4 - 416*K2**2*K3**2 - 304*K2**2*K4**2 + 824*K2**2*K4 - 1680*K2**2 - 32*K2*K3**2*K4 + 240*K2*K3*K5 + 192*K2*K4*K6 - 16*K3**4 + 24*K3**2*K6 - 852*K3**2 - 470*K4**2 - 48*K5**2 - 48*K6**2 + 1556 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.120'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11069', 'vk6.11147', 'vk6.12233', 'vk6.12340', 'vk6.18344', 'vk6.18681', 'vk6.24782', 'vk6.25239', 'vk6.30642', 'vk6.30735', 'vk6.31326', 'vk6.31727', 'vk6.31872', 'vk6.31941', 'vk6.32486', 'vk6.32897', 'vk6.36964', 'vk6.37421', 'vk6.39662', 'vk6.41901', 'vk6.44151', 'vk6.44471', 'vk6.46250', 'vk6.47855', 'vk6.51903', 'vk6.52328', 'vk6.52826', 'vk6.53172', 'vk6.56341', 'vk6.57620', 'vk6.60972', 'vk6.62292', 'vk6.63513', 'vk6.63557', 'vk6.63993', 'vk6.64037', 'vk6.65766', 'vk6.66027', 'vk6.68771', 'vk6.68979'] |
| 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 | O1O2O3O4O5O6U3U5U6U2U1U4 |
| R3 orbit | {'O1O2O3O4O5O6U3U5U6U2U1U4', 'O1O2O3O4O5U2O6U5U3U6U1U4'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5O6U3U6U5U1U2U4 |
| Gauss code of K* | O1O2O3O4O5O6U5U4U1U6U2U3 |
| Gauss code of -K* | O1O2O3O4O5O6U4U5U1U6U3U2 |
| 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 -3 3 0 2],[ 1 0 0 -2 3 0 2],[ 1 0 0 -2 2 0 2],[ 3 2 2 0 3 1 2],[-3 -3 -2 -3 0 -1 1],[ 0 0 0 -1 1 0 1],[-2 -2 -2 -2 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 2 0 -1 -1 -3],[-3 0 1 -1 -2 -3 -3],[-2 -1 0 -1 -2 -2 -2],[ 0 1 1 0 0 0 -1],[ 1 2 2 0 0 0 -2],[ 1 3 2 0 0 0 -2],[ 3 3 2 1 2 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,0,1,1,3,-1,1,2,3,3,1,2,2,2,0,0,1,0,2,2] |
| Phi over symmetry | [-3,-2,0,1,1,3,-1,1,2,3,3,1,2,2,2,0,0,1,0,2,2] |
| Phi of -K | [-3,-1,-1,0,2,3,0,0,2,3,3,0,1,1,1,1,1,2,1,2,2] |
| Phi of K* | [-3,-2,0,1,1,3,2,2,1,2,3,1,1,1,3,1,1,2,0,0,0] |
| Phi of -K* | [-3,-1,-1,0,2,3,2,2,1,2,3,0,0,2,2,0,2,3,1,1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 5z^2+18z+17 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+18w^2z+17w |
| Inner characteristic polynomial | t^6+46t^4+14t^2 |
| Outer characteristic polynomial | t^7+70t^5+51t^3+3t |
| Flat arrow polynomial | -2*K1**2 - 4*K1*K2 + 2*K1 + K2 + 2*K3 + 2 |
| 2-strand cable arrow polynomial | -1232*K1**2*K2**2 - 160*K1**2*K2*K4 + 1664*K1**2*K2 - 16*K1**2*K3**2 - 1716*K1**2 + 96*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 384*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 2224*K1*K2*K3 + 456*K1*K3*K4 + 8*K1*K5*K6 - 72*K2**4 - 416*K2**2*K3**2 - 304*K2**2*K4**2 + 824*K2**2*K4 - 1680*K2**2 - 32*K2*K3**2*K4 + 240*K2*K3*K5 + 192*K2*K4*K6 - 16*K3**4 + 24*K3**2*K6 - 852*K3**2 - 470*K4**2 - 48*K5**2 - 48*K6**2 + 1556 |
| 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}], [{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}], [{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}, {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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]] |
| If K is slice | False |