| Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,2,0,1,0,0,1,0,0,1,1,1,2,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1692'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.568', '6.806', '6.1000', '6.1049', '6.1081', '6.1101', '6.1112', '6.1122', '6.1193', '6.1195', '6.1208', '6.1235', '6.1263', '6.1517', '6.1528', '6.1537', '6.1542', '6.1545', '6.1558', '6.1569', '6.1575', '6.1644', '6.1650', '6.1681', '6.1692', '6.1702', '6.1706', '6.1728', '6.1734', '6.1739', '6.1799', '6.1813', '6.1820', '6.1834', '6.1840', '6.1851', '6.1861', '6.1878'] |
| Outer characteristic polynomial of the knot is: t^7+21t^5+27t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1692'] |
| 2-strand cable arrow polynomial of the knot is: -144*K1**4 + 192*K1**3*K2*K3 + 320*K1**2*K2**3 - 2368*K1**2*K2**2 - 512*K1**2*K2*K4 + 2536*K1**2*K2 - 208*K1**2*K3**2 - 2236*K1**2 + 32*K1*K2**3*K3 - 224*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 2968*K1*K2*K3 + 608*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 632*K2**4 - 16*K2**2*K3**2 - 48*K2**2*K4**2 + 872*K2**2*K4 - 1606*K2**2 + 16*K2*K3*K5 + 16*K2*K4*K6 - 936*K3**2 - 414*K4**2 - 4*K5**2 - 2*K6**2 + 1780 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1692'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19953', 'vk6.20097', 'vk6.21204', 'vk6.21377', 'vk6.26936', 'vk6.27166', 'vk6.28688', 'vk6.28852', 'vk6.38356', 'vk6.38572', 'vk6.40504', 'vk6.40765', 'vk6.45219', 'vk6.45463', 'vk6.47040', 'vk6.47202', 'vk6.56745', 'vk6.56918', 'vk6.57848', 'vk6.58052', 'vk6.61186', 'vk6.61457', 'vk6.62424', 'vk6.62607', 'vk6.66451', 'vk6.66622', 'vk6.67224', 'vk6.67409', 'vk6.69097', 'vk6.69266', 'vk6.69878', 'vk6.70004'] |
| 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 | O1O2O3U4O5U3U2U1O4O6U5U6 |
| R3 orbit | {'O1O2O3U4O5U3U2U1O4O6U5U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5O4O6U3U2U1O5U6 |
| Gauss code of K* | O1O2O3U4U5O6O5U3U2U1O4U6 |
| Gauss code of -K* | O1O2O3U4O5U3U2U1O6O4U6U5 |
| 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 0 0 0 -2 1 1],[ 0 0 0 0 -2 2 1],[ 0 0 0 0 -1 1 1],[ 0 0 0 0 0 0 1],[ 2 2 1 0 0 1 0],[-1 -2 -1 0 -1 0 1],[-1 -1 -1 -1 0 -1 0]] |
| Primitive based matrix | [[ 0 1 1 0 0 0 -2],[-1 0 1 0 -1 -2 -1],[-1 -1 0 -1 -1 -1 0],[ 0 0 1 0 0 0 0],[ 0 1 1 0 0 0 -1],[ 0 2 1 0 0 0 -2],[ 2 1 0 0 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,0,0,0,2,-1,0,1,2,1,1,1,1,0,0,0,0,0,1,2] |
| Phi over symmetry | [-2,0,0,0,1,1,0,1,2,0,1,0,0,1,0,0,1,1,1,2,-1] |
| Phi of -K | [-2,0,0,0,1,1,0,1,2,2,3,0,0,-1,0,0,0,0,1,0,-1] |
| Phi of K* | [-1,-1,0,0,0,2,-1,0,0,0,3,-1,0,1,2,0,0,0,0,1,2] |
| Phi of -K* | [-2,0,0,0,1,1,0,1,2,0,1,0,0,1,0,0,1,1,1,2,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | z^2+6z+9 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+5w^3z^2-12w^3z+18w^2z+9w |
| Inner characteristic polynomial | t^6+15t^4+14t^2 |
| Outer characteristic polynomial | t^7+21t^5+27t^3+6t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -144*K1**4 + 192*K1**3*K2*K3 + 320*K1**2*K2**3 - 2368*K1**2*K2**2 - 512*K1**2*K2*K4 + 2536*K1**2*K2 - 208*K1**2*K3**2 - 2236*K1**2 + 32*K1*K2**3*K3 - 224*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 2968*K1*K2*K3 + 608*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 632*K2**4 - 16*K2**2*K3**2 - 48*K2**2*K4**2 + 872*K2**2*K4 - 1606*K2**2 + 16*K2*K3*K5 + 16*K2*K4*K6 - 936*K3**2 - 414*K4**2 - 4*K5**2 - 2*K6**2 + 1780 |
| 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}], [{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}, {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}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {1, 3}, {2}]] |
| If K is slice | False |