| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,2,3,1,0,1,1,0,0,0,-1,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1662'] |
| Arrow polynomial of the knot is: 4*K1**3 - 14*K1**2 - 4*K1*K2 - K1 + 7*K2 + K3 + 8 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1023', '6.1530', '6.1662', '6.1668', '6.1801'] |
| Outer characteristic polynomial of the knot is: t^7+28t^5+43t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.374', '6.1511', '6.1662'] |
| 2-strand cable arrow polynomial of the knot is: -704*K1**6 - 576*K1**4*K2**2 + 2880*K1**4*K2 - 6336*K1**4 + 960*K1**3*K2*K3 - 704*K1**3*K3 - 192*K1**2*K2**4 + 1248*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 9696*K1**2*K2**2 - 480*K1**2*K2*K4 + 13656*K1**2*K2 - 1024*K1**2*K3**2 - 80*K1**2*K4**2 - 5716*K1**2 + 256*K1*K2**3*K3 - 1952*K1*K2**2*K3 - 64*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 9528*K1*K2*K3 + 1528*K1*K3*K4 + 128*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1128*K2**4 - 256*K2**2*K3**2 - 48*K2**2*K4**2 + 1592*K2**2*K4 - 5574*K2**2 + 240*K2*K3*K5 + 16*K2*K4*K6 - 2516*K3**2 - 658*K4**2 - 64*K5**2 - 2*K6**2 + 5768 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1662'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4142', 'vk6.4173', 'vk6.5380', 'vk6.5411', 'vk6.7502', 'vk6.7529', 'vk6.9003', 'vk6.9034', 'vk6.12442', 'vk6.12475', 'vk6.13338', 'vk6.13563', 'vk6.13594', 'vk6.14250', 'vk6.14699', 'vk6.14739', 'vk6.15206', 'vk6.15853', 'vk6.15893', 'vk6.30855', 'vk6.30888', 'vk6.32039', 'vk6.32072', 'vk6.33056', 'vk6.33087', 'vk6.33865', 'vk6.34324', 'vk6.48498', 'vk6.50277', 'vk6.53526', 'vk6.53938', 'vk6.54258'] |
| 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 | O1O2O3U2O4U1U4U5O6O5U3U6 |
| R3 orbit | {'O1O2O3U2O4U1U4U5O6O5U3U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U1O5O4U5U6U3O6U2 |
| Gauss code of K* | O1O2O3U4U3O5O4U1U6U5O6U2 |
| Gauss code of -K* | O1O2O3U2O4U5U4U3O6O5U1U6 |
| 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 1 1 0],[ 2 0 0 3 1 2 1],[ 1 0 0 1 0 1 1],[-1 -3 -1 0 0 -1 0],[-1 -1 0 0 0 -1 0],[-1 -2 -1 1 1 0 0],[ 0 -1 -1 0 0 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 1 0 -1 -2],[-1 -1 0 0 0 0 -1],[-1 -1 0 0 0 -1 -3],[ 0 0 0 0 0 -1 -1],[ 1 1 0 1 1 0 0],[ 2 2 1 3 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,-1,0,1,2,0,0,0,1,0,1,3,1,1,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,1,1,2,3,1,0,1,1,0,0,0,-1,0,1] |
| Phi of -K | [-2,-1,0,1,1,1,1,1,0,1,2,0,1,1,2,1,1,1,1,0,-1] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,1,1,0,1,1,1,1,1,2,2,0,1,1] |
| Phi of -K* | [-2,-1,0,1,1,1,0,1,1,2,3,1,0,1,1,0,0,0,-1,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 2z^2+23z+39 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2+23w^2z+39w |
| Inner characteristic polynomial | t^6+20t^4+18t^2+1 |
| Outer characteristic polynomial | t^7+28t^5+43t^3+8t |
| Flat arrow polynomial | 4*K1**3 - 14*K1**2 - 4*K1*K2 - K1 + 7*K2 + K3 + 8 |
| 2-strand cable arrow polynomial | -704*K1**6 - 576*K1**4*K2**2 + 2880*K1**4*K2 - 6336*K1**4 + 960*K1**3*K2*K3 - 704*K1**3*K3 - 192*K1**2*K2**4 + 1248*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 9696*K1**2*K2**2 - 480*K1**2*K2*K4 + 13656*K1**2*K2 - 1024*K1**2*K3**2 - 80*K1**2*K4**2 - 5716*K1**2 + 256*K1*K2**3*K3 - 1952*K1*K2**2*K3 - 64*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 9528*K1*K2*K3 + 1528*K1*K3*K4 + 128*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1128*K2**4 - 256*K2**2*K3**2 - 48*K2**2*K4**2 + 1592*K2**2*K4 - 5574*K2**2 + 240*K2*K3*K5 + 16*K2*K4*K6 - 2516*K3**2 - 658*K4**2 - 64*K5**2 - 2*K6**2 + 5768 |
| 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}, {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}, {3}, {1, 2}]] |
| If K is slice | False |