| Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,0,2,2,2,4,0,2,1,1,1,2,1,1,1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.586'] |
| Arrow polynomial of the knot is: -4*K1**2 - 6*K1*K2 + 3*K1 + 2*K2 + 3*K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.586', '6.590', '6.958', '6.987', '6.991', '6.993', '6.999', '6.1054', '6.1065', '6.1096', '6.1168', '6.1182'] |
| Outer characteristic polynomial of the knot is: t^7+49t^5+105t^3+16t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.586'] |
| 2-strand cable arrow polynomial of the knot is: -96*K1**4 + 128*K1**3*K2*K3 - 32*K1**3*K3 - 64*K1**2*K2**2*K3**2 - 2896*K1**2*K2**2 + 96*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 32*K1**2*K2*K4 + 2944*K1**2*K2 - 560*K1**2*K3**2 - 3580*K1**2 + 576*K1*K2**3*K3 - 448*K1*K2**2*K3 - 32*K1*K2**2*K5 + 64*K1*K2*K3**3 - 224*K1*K2*K3*K4 + 6248*K1*K2*K3 - 32*K1*K3**2*K5 + 768*K1*K3*K4 + 40*K1*K4*K5 - 544*K2**4 - 1088*K2**2*K3**2 - 24*K2**2*K4**2 + 544*K2**2*K4 - 2802*K2**2 + 720*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 8*K3**2*K6 - 2396*K3**2 - 340*K4**2 - 128*K5**2 - 14*K6**2 + 3138 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.586'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71480', 'vk6.71504', 'vk6.71539', 'vk6.71561', 'vk6.72014', 'vk6.72030', 'vk6.72065', 'vk6.72080', 'vk6.72517', 'vk6.72533', 'vk6.72654', 'vk6.72656', 'vk6.72913', 'vk6.72951', 'vk6.73127', 'vk6.73130', 'vk6.73654', 'vk6.73695', 'vk6.73699', 'vk6.77101', 'vk6.77122', 'vk6.77153', 'vk6.77175', 'vk6.77446', 'vk6.77468', 'vk6.77954', 'vk6.77957', 'vk6.78589', 'vk6.81426', 'vk6.86907', 'vk6.87253', 'vk6.89344'] |
| 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 | O1O2O3O4U1O5U4O6U5U2U6U3 |
| R3 orbit | {'O1O2O3O4U1O5U4O6U5U2U6U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U2U5U3U6O5U1O6U4 |
| Gauss code of K* | O1O2O3O4U5U2U4U6O5U1O6U3 |
| Gauss code of -K* | O1O2O3O4U2O5U4O6U5U1U3U6 |
| 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 -1 2 0 0 2],[ 3 0 2 3 1 1 1],[ 1 -2 0 2 -1 1 2],[-2 -3 -2 0 -1 0 1],[ 0 -1 1 1 0 1 1],[ 0 -1 -1 0 -1 0 1],[-2 -1 -2 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 2 0 0 -1 -3],[-2 0 1 0 -1 -2 -3],[-2 -1 0 -1 -1 -2 -1],[ 0 0 1 0 -1 -1 -1],[ 0 1 1 1 0 1 -1],[ 1 2 2 1 -1 0 -2],[ 3 3 1 1 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,0,0,1,3,-1,0,1,2,3,1,1,2,1,1,1,1,-1,1,2] |
| Phi over symmetry | [-3,-1,0,0,2,2,0,2,2,2,4,0,2,1,1,1,2,1,1,1,-1] |
| Phi of -K | [-3,-1,0,0,2,2,0,2,2,2,4,0,2,1,1,1,2,1,1,1,-1] |
| Phi of K* | [-2,-2,0,0,1,3,-1,1,1,1,4,1,2,1,2,1,2,2,0,2,0] |
| Phi of -K* | [-3,-1,0,0,2,2,2,1,1,1,3,-1,1,2,2,1,1,1,1,0,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-2t^2+t |
| Normalized Jones-Krushkal polynomial | 2z^2+15z+23 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2-8w^3z+23w^2z+23w |
| Inner characteristic polynomial | t^6+31t^4+29t^2+1 |
| Outer characteristic polynomial | t^7+49t^5+105t^3+16t |
| Flat arrow polynomial | -4*K1**2 - 6*K1*K2 + 3*K1 + 2*K2 + 3*K3 + 3 |
| 2-strand cable arrow polynomial | -96*K1**4 + 128*K1**3*K2*K3 - 32*K1**3*K3 - 64*K1**2*K2**2*K3**2 - 2896*K1**2*K2**2 + 96*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 32*K1**2*K2*K4 + 2944*K1**2*K2 - 560*K1**2*K3**2 - 3580*K1**2 + 576*K1*K2**3*K3 - 448*K1*K2**2*K3 - 32*K1*K2**2*K5 + 64*K1*K2*K3**3 - 224*K1*K2*K3*K4 + 6248*K1*K2*K3 - 32*K1*K3**2*K5 + 768*K1*K3*K4 + 40*K1*K4*K5 - 544*K2**4 - 1088*K2**2*K3**2 - 24*K2**2*K4**2 + 544*K2**2*K4 - 2802*K2**2 + 720*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 8*K3**2*K6 - 2396*K3**2 - 340*K4**2 - 128*K5**2 - 14*K6**2 + 3138 |
| 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}], [{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}, {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}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}]] |
| If K is slice | False |