| Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,2,1,2,3,1,0,1,1,1,1,2,0,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.552'] |
| Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 6*K1*K2 + 2*K2 + 2*K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.552', '6.652', '6.764', '6.776', '6.784', '6.839', '6.903', '6.1010', '6.1166'] |
| Outer characteristic polynomial of the knot is: t^7+68t^5+62t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.552'] |
| 2-strand cable arrow polynomial of the knot is: -672*K1**4 - 384*K1**3*K3 + 128*K1**2*K2**3 - 1200*K1**2*K2**2 - 256*K1**2*K2*K4 + 2952*K1**2*K2 - 512*K1**2*K3**2 - 2472*K1**2 + 96*K1*K2**3*K3 + 32*K1*K2*K3**3 - 224*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 2904*K1*K2*K3 - 64*K1*K3**2*K5 + 664*K1*K3*K4 + 48*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 272*K2**4 - 272*K2**2*K3**2 - 56*K2**2*K4**2 + 416*K2**2*K4 - 1796*K2**2 + 392*K2*K3*K5 + 40*K2*K4*K6 - 96*K3**4 + 128*K3**2*K6 - 1076*K3**2 - 284*K4**2 - 92*K5**2 - 36*K6**2 + 1954 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.552'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4418', 'vk6.4513', 'vk6.5800', 'vk6.5927', 'vk6.7869', 'vk6.7976', 'vk6.9287', 'vk6.9406', 'vk6.10168', 'vk6.10241', 'vk6.10386', 'vk6.17869', 'vk6.17932', 'vk6.18296', 'vk6.18634', 'vk6.24376', 'vk6.24676', 'vk6.25184', 'vk6.30063', 'vk6.30126', 'vk6.30905', 'vk6.31030', 'vk6.32093', 'vk6.32214', 'vk6.36907', 'vk6.37262', 'vk6.37367', 'vk6.43811', 'vk6.44124', 'vk6.44449', 'vk6.50516', 'vk6.50597', 'vk6.51118', 'vk6.51997', 'vk6.52094', 'vk6.55836', 'vk6.56082', 'vk6.60562', 'vk6.60901', 'vk6.65971'] |
| 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 | O1O2O3O4O5U3U6U4O6U1U5U2 |
| R3 orbit | {'O1O2O3O4O5U3U6U4O6U1U5U2', 'O1O2O3O4U2O5U6U4O6U1U3U5'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U4U1U5O6U2U6U3 |
| Gauss code of K* | O1O2O3U2O4O5O6U4U6U1U3U5 |
| Gauss code of -K* | O1O2O3U4O5O4O6U2U5U6U1U3 |
| 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 -2 1 3 -1],[ 2 0 2 -1 2 3 1],[-1 -2 0 -2 1 2 -2],[ 2 1 2 0 1 2 1],[-1 -2 -1 -1 0 0 -1],[-3 -3 -2 -2 0 0 -3],[ 1 -1 2 -1 1 3 0]] |
| Primitive based matrix | [[ 0 3 1 1 -1 -2 -2],[-3 0 0 -2 -3 -2 -3],[-1 0 0 -1 -1 -1 -2],[-1 2 1 0 -2 -2 -2],[ 1 3 1 2 0 -1 -1],[ 2 2 1 2 1 0 1],[ 2 3 2 2 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,-1,1,2,2,0,2,3,2,3,1,1,1,2,2,2,2,1,1,-1] |
| Phi over symmetry | [-3,-1,-1,1,2,2,0,2,1,2,3,1,0,1,1,1,1,2,0,0,-1] |
| Phi of -K | [-2,-2,-1,1,1,3,-1,0,1,2,3,0,1,1,2,0,1,1,-1,0,2] |
| Phi of K* | [-3,-1,-1,1,2,2,0,2,1,2,3,1,0,1,1,1,1,2,0,0,-1] |
| Phi of -K* | [-2,-2,-1,1,1,3,-1,1,2,2,3,1,1,2,2,1,2,3,-1,0,2] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+2t^2-t |
| Normalized Jones-Krushkal polynomial | z^2+14z+25 |
| Enhanced Jones-Krushkal polynomial | w^3z^2+14w^2z+25w |
| Inner characteristic polynomial | t^6+48t^4+36t^2 |
| Outer characteristic polynomial | t^7+68t^5+62t^3+3t |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 6*K1*K2 + 2*K2 + 2*K3 + 3 |
| 2-strand cable arrow polynomial | -672*K1**4 - 384*K1**3*K3 + 128*K1**2*K2**3 - 1200*K1**2*K2**2 - 256*K1**2*K2*K4 + 2952*K1**2*K2 - 512*K1**2*K3**2 - 2472*K1**2 + 96*K1*K2**3*K3 + 32*K1*K2*K3**3 - 224*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 2904*K1*K2*K3 - 64*K1*K3**2*K5 + 664*K1*K3*K4 + 48*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 272*K2**4 - 272*K2**2*K3**2 - 56*K2**2*K4**2 + 416*K2**2*K4 - 1796*K2**2 + 392*K2*K3*K5 + 40*K2*K4*K6 - 96*K3**4 + 128*K3**2*K6 - 1076*K3**2 - 284*K4**2 - 92*K5**2 - 36*K6**2 + 1954 |
| 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}], [{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}], [{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 |