| Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,1,3,2,4,0,1,1,2,0,1,1,2,1,-2] |
| Flat knots (up to 7 crossings) with same phi are :['6.595'] |
| 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+69t^5+78t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.595'] |
| 2-strand cable arrow polynomial of the knot is: 32*K1**4*K2 - 1024*K1**4 - 224*K1**3*K3 + 32*K1**2*K2**3 - 1776*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 128*K1**2*K2*K4 + 6656*K1**2*K2 - 224*K1**2*K3**2 - 6188*K1**2 + 128*K1*K2**3*K3 - 1120*K1*K2**2*K3 - 96*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 5144*K1*K2*K3 + 920*K1*K3*K4 - 376*K2**4 - 224*K2**2*K3**2 - 8*K2**2*K4**2 + 1152*K2**2*K4 - 4742*K2**2 + 248*K2*K3*K5 + 8*K2*K4*K6 - 2100*K3**2 - 570*K4**2 - 56*K5**2 - 2*K6**2 + 4536 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.595'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72417', 'vk6.72433', 'vk6.72466', 'vk6.72489', 'vk6.72495', 'vk6.72496', 'vk6.72831', 'vk6.72848', 'vk6.72857', 'vk6.72858', 'vk6.72892', 'vk6.72907', 'vk6.74451', 'vk6.74456', 'vk6.74468', 'vk6.74477', 'vk6.75066', 'vk6.75069', 'vk6.76961', 'vk6.77777', 'vk6.77790', 'vk6.77970', 'vk6.79456', 'vk6.79459', 'vk6.79904', 'vk6.79909', 'vk6.79924', 'vk6.79927', 'vk6.80926', 'vk6.80940', 'vk6.87237', 'vk6.89362'] |
| 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 | O1O2O3O4U2O5U1O6U5U4U6U3 |
| R3 orbit | {'O1O2O3O4U2O5U1O6U5U4U6U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U2U5U1U6O5U4O6U3 |
| Gauss code of K* | O1O2O3O4U5U6U4U2O6U1O5U3 |
| Gauss code of -K* | O1O2O3O4U2O5U4O6U3U1U6U5 |
| 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 -3 -2 2 1 0 2],[ 3 0 0 4 3 1 2],[ 2 0 0 2 1 0 1],[-2 -4 -2 0 -1 -1 2],[-1 -3 -1 1 0 0 2],[ 0 -1 0 1 0 0 1],[-2 -2 -1 -2 -2 -1 0]] |
| Primitive based matrix | [[ 0 2 2 1 0 -2 -3],[-2 0 2 -1 -1 -2 -4],[-2 -2 0 -2 -1 -1 -2],[-1 1 2 0 0 -1 -3],[ 0 1 1 0 0 0 -1],[ 2 2 1 1 0 0 0],[ 3 4 2 3 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-1,0,2,3,-2,1,1,2,4,2,1,1,2,0,1,3,0,1,0] |
| Phi over symmetry | [-3,-2,0,1,2,2,0,1,3,2,4,0,1,1,2,0,1,1,2,1,-2] |
| Phi of -K | [-3,-2,0,1,2,2,1,2,1,1,3,2,2,2,3,1,1,1,0,-1,-2] |
| Phi of K* | [-2,-2,-1,0,2,3,-2,-1,1,3,3,0,1,2,1,1,2,1,2,2,1] |
| Phi of -K* | [-3,-2,0,1,2,2,0,1,3,2,4,0,1,1,2,0,1,1,2,1,-2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 2z^2+21z+35 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2+21w^2z+35w |
| Inner characteristic polynomial | t^6+47t^4+21t^2 |
| Outer characteristic polynomial | t^7+69t^5+78t^3+5t |
| Flat arrow polynomial | -6*K1**2 - 2*K1*K2 + K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | 32*K1**4*K2 - 1024*K1**4 - 224*K1**3*K3 + 32*K1**2*K2**3 - 1776*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 128*K1**2*K2*K4 + 6656*K1**2*K2 - 224*K1**2*K3**2 - 6188*K1**2 + 128*K1*K2**3*K3 - 1120*K1*K2**2*K3 - 96*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 5144*K1*K2*K3 + 920*K1*K3*K4 - 376*K2**4 - 224*K2**2*K3**2 - 8*K2**2*K4**2 + 1152*K2**2*K4 - 4742*K2**2 + 248*K2*K3*K5 + 8*K2*K4*K6 - 2100*K3**2 - 570*K4**2 - 56*K5**2 - 2*K6**2 + 4536 |
| 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}], [{1, 6}, {5}, {4}, {3}, {2}], [{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}], [{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}, {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}, {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 |