| Min(phi) over symmetries of the knot is: [-3,0,0,0,1,2,1,1,2,2,2,0,0,0,0,0,0,1,0,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1059'] |
| Arrow polynomial of the knot is: 8*K1**3 - 6*K1**2 - 6*K1*K2 - 3*K1 + 3*K2 + K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.237', '6.602', '6.956', '6.986', '6.992', '6.1052', '6.1059'] |
| Outer characteristic polynomial of the knot is: t^7+31t^5+20t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1059'] |
| 2-strand cable arrow polynomial of the knot is: -784*K1**4 + 192*K1**3*K2*K3 - 416*K1**3*K3 + 352*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 2736*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 4056*K1**2*K2 - 432*K1**2*K3**2 - 32*K1**2*K3*K5 - 2752*K1**2 + 608*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 448*K1*K2**2*K3 - 64*K1*K2**2*K5 + 32*K1*K2*K3**3 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 3608*K1*K2*K3 + 512*K1*K3*K4 + 24*K1*K4*K5 - 64*K2**6 + 96*K2**4*K4 - 856*K2**4 - 560*K2**2*K3**2 - 72*K2**2*K4**2 + 672*K2**2*K4 - 1614*K2**2 + 288*K2*K3*K5 + 24*K2*K4*K6 - 972*K3**2 - 178*K4**2 - 28*K5**2 - 2*K6**2 + 1992 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1059'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71353', 'vk6.71403', 'vk6.71409', 'vk6.71868', 'vk6.71870', 'vk6.71929', 'vk6.71931', 'vk6.74194', 'vk6.74328', 'vk6.74337', 'vk6.74973', 'vk6.74982', 'vk6.75622', 'vk6.75810', 'vk6.76359', 'vk6.76544', 'vk6.76549', 'vk6.76950', 'vk6.76999', 'vk6.77013', 'vk6.77058', 'vk6.78605', 'vk6.78803', 'vk6.79223', 'vk6.79373', 'vk6.79798', 'vk6.79807', 'vk6.80247', 'vk6.80700', 'vk6.80838', 'vk6.81269', 'vk6.81465', 'vk6.84058', 'vk6.86008', 'vk6.87063', 'vk6.87081', 'vk6.87743', 'vk6.88035', 'vk6.88202', 'vk6.89397'] |
| 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 | O1O2O3O4U5U1O6U3O5U6U2U4 |
| R3 orbit | {'O1O2O3O4U5U1U2O6O5U3U6U4', 'O1O2O3O4U5U1O6U3O5U6U2U4'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U1U3U5O6U2O5U4U6 |
| Gauss code of K* | O1O2O3U4U2U5U3O6O4U1O5U6 |
| Gauss code of -K* | O1O2O3U4O5U3O6O4U1U5U2U6 |
| 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 0 0 3 -1 0],[ 2 0 1 1 2 1 0],[ 0 -1 0 0 2 0 0],[ 0 -1 0 0 1 0 0],[-3 -2 -2 -1 0 -2 -1],[ 1 -1 0 0 2 0 0],[ 0 0 0 0 1 0 0]] |
| Primitive based matrix | [[ 0 3 0 0 0 -1 -2],[-3 0 -1 -1 -2 -2 -2],[ 0 1 0 0 0 0 0],[ 0 1 0 0 0 0 -1],[ 0 2 0 0 0 0 -1],[ 1 2 0 0 0 0 -1],[ 2 2 0 1 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,0,0,0,1,2,1,1,2,2,2,0,0,0,0,0,0,1,0,1,1] |
| Phi over symmetry | [-3,0,0,0,1,2,1,1,2,2,2,0,0,0,0,0,0,1,0,1,1] |
| Phi of -K | [-2,-1,0,0,0,3,0,1,1,2,3,1,1,1,2,0,0,1,0,2,2] |
| Phi of K* | [-3,0,0,0,1,2,1,2,2,2,3,0,0,1,1,0,1,1,1,2,0] |
| Phi of -K* | [-2,-1,0,0,0,3,1,0,1,1,2,0,0,0,2,0,0,1,0,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 2z^2+15z+23 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2+15w^2z+23w |
| Inner characteristic polynomial | t^6+17t^4+5t^2 |
| Outer characteristic polynomial | t^7+31t^5+20t^3+3t |
| Flat arrow polynomial | 8*K1**3 - 6*K1**2 - 6*K1*K2 - 3*K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -784*K1**4 + 192*K1**3*K2*K3 - 416*K1**3*K3 + 352*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 2736*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 4056*K1**2*K2 - 432*K1**2*K3**2 - 32*K1**2*K3*K5 - 2752*K1**2 + 608*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 448*K1*K2**2*K3 - 64*K1*K2**2*K5 + 32*K1*K2*K3**3 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 3608*K1*K2*K3 + 512*K1*K3*K4 + 24*K1*K4*K5 - 64*K2**6 + 96*K2**4*K4 - 856*K2**4 - 560*K2**2*K3**2 - 72*K2**2*K4**2 + 672*K2**2*K4 - 1614*K2**2 + 288*K2*K3*K5 + 24*K2*K4*K6 - 972*K3**2 - 178*K4**2 - 28*K5**2 - 2*K6**2 + 1992 |
| 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 |