| Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,3,1,2,3,2,0,2,3,1,1,1,1,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.616'] |
| Arrow polynomial of the knot is: -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.217', '6.219', '6.304', '6.349', '6.390', '6.400', '6.416', '6.515', '6.518', '6.530', '6.582', '6.616', '6.629', '6.641', '6.645', '6.702', '6.710', '6.715', '6.729', '6.733', '6.734', '6.802', '6.840', '6.845', '6.854', '6.860', '6.900', '6.905', '6.921', '6.924', '6.979', '6.980', '6.996', '6.1044', '6.1067', '6.1086', '6.1100', '6.1139', '6.1145', '6.1149', '6.1167', '6.1169', '6.1183', '6.1314'] |
| Outer characteristic polynomial of the knot is: t^7+67t^5+235t^3+12t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.616'] |
| 2-strand cable arrow polynomial of the knot is: 32*K1**4*K2 - 256*K1**4 - 64*K1**3*K3 + 640*K1**2*K2**3 + 96*K1**2*K2**2*K4 - 3552*K1**2*K2**2 - 704*K1**2*K2*K4 + 4984*K1**2*K2 - 384*K1**2*K3**2 - 64*K1**2*K4**2 - 4704*K1**2 + 384*K1*K2**3*K3 - 1344*K1*K2**2*K3 - 352*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 5624*K1*K2*K3 + 1704*K1*K3*K4 + 216*K1*K4*K5 - 888*K2**4 - 320*K2**2*K3**2 - 8*K2**2*K4**2 + 1840*K2**2*K4 - 3630*K2**2 + 448*K2*K3*K5 + 8*K2*K4*K6 - 2056*K3**2 - 1042*K4**2 - 168*K5**2 - 2*K6**2 + 3720 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.616'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71581', 'vk6.71702', 'vk6.72120', 'vk6.72320', 'vk6.73473', 'vk6.74124', 'vk6.74133', 'vk6.74693', 'vk6.74703', 'vk6.75227', 'vk6.75479', 'vk6.76165', 'vk6.76178', 'vk6.77199', 'vk6.77304', 'vk6.77507', 'vk6.77659', 'vk6.78443', 'vk6.79121', 'vk6.79136', 'vk6.80027', 'vk6.80177', 'vk6.80629', 'vk6.80637', 'vk6.83737', 'vk6.83860', 'vk6.85061', 'vk6.85339', 'vk6.86663', 'vk6.86985', 'vk6.87415', 'vk6.89545'] |
| 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 | O1O2O3O4U5O6U1O5U2U6U4U3 |
| R3 orbit | {'O1O2O3O4U5O6U1O5U2U6U4U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U2U1U5U3O6U4O5U6 |
| Gauss code of K* | O1O2O3O4U5U1U4U3O6U2O5U6 |
| Gauss code of -K* | O1O2O3O4U5O6U3O5U2U1U4U6 |
| 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 2 0 1],[ 3 0 0 3 2 3 1],[ 2 0 0 3 2 2 0],[-2 -3 -3 0 0 -1 -1],[-2 -2 -2 0 0 -1 -1],[ 0 -3 -2 1 1 0 1],[-1 -1 0 1 1 -1 0]] |
| Primitive based matrix | [[ 0 2 2 1 0 -2 -3],[-2 0 0 -1 -1 -2 -2],[-2 0 0 -1 -1 -3 -3],[-1 1 1 0 -1 0 -1],[ 0 1 1 1 0 -2 -3],[ 2 2 3 0 2 0 0],[ 3 2 3 1 3 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-1,0,2,3,0,1,1,2,2,1,1,3,3,1,0,1,2,3,0] |
| Phi over symmetry | [-3,-2,0,1,2,2,0,3,1,2,3,2,0,2,3,1,1,1,1,1,0] |
| Phi of -K | [-3,-2,0,1,2,2,1,0,3,2,3,0,3,1,2,0,1,1,0,0,0] |
| Phi of K* | [-2,-2,-1,0,2,3,0,0,1,1,2,0,1,2,3,0,3,3,0,0,1] |
| Phi of -K* | [-3,-2,0,1,2,2,0,3,1,2,3,2,0,2,3,1,1,1,1,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 6z^2+23z+23 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2-4w^3z+27w^2z+23w |
| Inner characteristic polynomial | t^6+45t^4+116t^2 |
| Outer characteristic polynomial | t^7+67t^5+235t^3+12t |
| Flat arrow polynomial | -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | 32*K1**4*K2 - 256*K1**4 - 64*K1**3*K3 + 640*K1**2*K2**3 + 96*K1**2*K2**2*K4 - 3552*K1**2*K2**2 - 704*K1**2*K2*K4 + 4984*K1**2*K2 - 384*K1**2*K3**2 - 64*K1**2*K4**2 - 4704*K1**2 + 384*K1*K2**3*K3 - 1344*K1*K2**2*K3 - 352*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 5624*K1*K2*K3 + 1704*K1*K3*K4 + 216*K1*K4*K5 - 888*K2**4 - 320*K2**2*K3**2 - 8*K2**2*K4**2 + 1840*K2**2*K4 - 3630*K2**2 + 448*K2*K3*K5 + 8*K2*K4*K6 - 2056*K3**2 - 1042*K4**2 - 168*K5**2 - 2*K6**2 + 3720 |
| 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}], [{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}, {2, 5}, {4}, {3}, {1}], [{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}, {3, 4}, {1, 2}]] |
| If K is slice | False |