| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,1,2,3,1,1,1,1,0,0,1,-1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.797'] |
| Arrow polynomial of the knot is: -6*K1**2 - 2*K1*K2 + K1 + 3*K2 + K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.323', '6.380', '6.444', '6.472', '6.523', '6.579', '6.592', '6.595', '6.609', '6.614', '6.620', '6.644', '6.648', '6.669', '6.671', '6.681', '6.693', '6.724', '6.725', '6.757', '6.766', '6.785', '6.786', '6.797', '6.798', '6.816', '6.833', '6.972', '6.978', '6.1056', '6.1064', '6.1066', '6.1087', '6.1094', '6.1273', '6.1277', '6.1282', '6.1295', '6.1300', '6.1313', '6.1344', '6.1353', '6.1354'] |
| Outer characteristic polynomial of the knot is: t^7+40t^5+64t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.410', '6.797', '7.19581'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 + 64*K1**4*K2 - 2592*K1**4 - 256*K1**3*K3 - 2288*K1**2*K2**2 - 288*K1**2*K2*K4 + 5608*K1**2*K2 - 992*K1**2*K3**2 - 192*K1**2*K4**2 - 3748*K1**2 - 192*K1*K2**2*K3 + 5048*K1*K2*K3 + 1816*K1*K3*K4 + 208*K1*K4*K5 - 88*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 344*K2**2*K4 - 3126*K2**2 + 64*K2*K3*K5 + 8*K2*K4*K6 - 2072*K3**2 - 698*K4**2 - 76*K5**2 - 2*K6**2 + 3568 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.797'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11523', 'vk6.11855', 'vk6.12870', 'vk6.13178', 'vk6.20361', 'vk6.21702', 'vk6.27661', 'vk6.29205', 'vk6.31292', 'vk6.31687', 'vk6.32446', 'vk6.32861', 'vk6.39103', 'vk6.41357', 'vk6.45855', 'vk6.47516', 'vk6.52308', 'vk6.52571', 'vk6.53148', 'vk6.53452', 'vk6.57220', 'vk6.58441', 'vk6.61830', 'vk6.62961', 'vk6.63809', 'vk6.63942', 'vk6.64252', 'vk6.64449', 'vk6.66833', 'vk6.67701', 'vk6.69469', 'vk6.70191'] |
| 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 | O1O2O3O4U5O6U3U6U1O5U2U4 |
| R3 orbit | {'O1O2O3O4U5O6U3U6U1O5U2U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U3O5U4U6U2O6U5 |
| Gauss code of K* | O1O2O3U4O5O6U3U5U1U6O4U2 |
| Gauss code of -K* | O1O2O3U2O4U5U3U6U1O5O6U4 |
| 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 0 -1 3 -2 1],[ 1 0 0 -1 2 0 1],[ 0 0 0 0 2 -1 1],[ 1 1 0 0 1 0 1],[-3 -2 -2 -1 0 -3 0],[ 2 0 1 0 3 0 1],[-1 -1 -1 -1 0 -1 0]] |
| Primitive based matrix | [[ 0 3 1 0 -1 -1 -2],[-3 0 0 -2 -1 -2 -3],[-1 0 0 -1 -1 -1 -1],[ 0 2 1 0 0 0 -1],[ 1 1 1 0 0 1 0],[ 1 2 1 0 -1 0 0],[ 2 3 1 1 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,1,1,2,0,2,1,2,3,1,1,1,1,0,0,1,-1,0,0] |
| Phi over symmetry | [-3,-1,0,1,1,2,0,2,1,2,3,1,1,1,1,0,0,1,-1,0,0] |
| Phi of -K | [-2,-1,-1,0,1,3,1,1,1,2,2,-1,1,1,3,1,1,2,0,1,2] |
| Phi of K* | [-3,-1,0,1,1,2,2,1,2,3,2,0,1,1,2,1,1,1,-1,1,1] |
| Phi of -K* | [-2,-1,-1,0,1,3,0,0,1,1,3,-1,0,1,2,0,1,1,1,2,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 4z^2+25z+35 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+25w^2z+35w |
| Inner characteristic polynomial | t^6+24t^4+31t^2 |
| Outer characteristic polynomial | t^7+40t^5+64t^3+7t |
| Flat arrow polynomial | -6*K1**2 - 2*K1*K2 + K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -64*K1**6 + 64*K1**4*K2 - 2592*K1**4 - 256*K1**3*K3 - 2288*K1**2*K2**2 - 288*K1**2*K2*K4 + 5608*K1**2*K2 - 992*K1**2*K3**2 - 192*K1**2*K4**2 - 3748*K1**2 - 192*K1*K2**2*K3 + 5048*K1*K2*K3 + 1816*K1*K3*K4 + 208*K1*K4*K5 - 88*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 344*K2**2*K4 - 3126*K2**2 + 64*K2*K3*K5 + 8*K2*K4*K6 - 2072*K3**2 - 698*K4**2 - 76*K5**2 - 2*K6**2 + 3568 |
| 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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]] |
| If K is slice | False |