| Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,1,1,2,2,1,1,2,2,1,1,1,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.868'] |
| Arrow polynomial of the knot is: -16*K1**4 + 4*K1**3 + 8*K1**2*K2 + 6*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.193', '6.868'] |
| Outer characteristic polynomial of the knot is: t^7+77t^5+56t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.868'] |
| 2-strand cable arrow polynomial of the knot is: -864*K1**4 - 1792*K1**2*K2**6 + 2176*K1**2*K2**5 - 3264*K1**2*K2**4 + 1184*K1**2*K2**3 - 2048*K1**2*K2**2 + 2512*K1**2*K2 - 192*K1**2*K3**2 - 1252*K1**2 + 1408*K1*K2**5*K3 + 1088*K1*K2**3*K3 + 1696*K1*K2*K3 + 224*K1*K3*K4 - 768*K2**8 + 512*K2**6*K4 - 1568*K2**6 - 320*K2**4*K3**2 - 64*K2**4*K4**2 + 544*K2**4*K4 + 520*K2**4 + 32*K2**3*K3*K5 - 240*K2**2*K3**2 - 40*K2**2*K4**2 + 376*K2**2*K4 - 520*K2**2 + 112*K2*K3*K5 - 440*K3**2 - 110*K4**2 - 20*K5**2 + 1116 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.868'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16901', 'vk6.17143', 'vk6.20202', 'vk6.21482', 'vk6.23289', 'vk6.23588', 'vk6.27372', 'vk6.29008', 'vk6.35295', 'vk6.35733', 'vk6.38803', 'vk6.40976', 'vk6.42806', 'vk6.43088', 'vk6.45554', 'vk6.47341', 'vk6.55050', 'vk6.55293', 'vk6.57041', 'vk6.58135', 'vk6.59442', 'vk6.59729', 'vk6.61527', 'vk6.62714', 'vk6.64893', 'vk6.65106', 'vk6.66655', 'vk6.67476', 'vk6.68202', 'vk6.68346', 'vk6.69301', 'vk6.70066'] |
| 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 | O1O2O3O4U1U5O6O5U2U3U4U6 |
| R3 orbit | {'O1O2O3O4U1U5O6O5U2U3U4U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U1U2U3O6O5U6U4 |
| Gauss code of K* | O1O2O3O4U5U1U2U3O5O6U4U6 |
| Gauss code of -K* | O1O2O3O4U5U1O5O6U2U3U4U6 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 2 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -3 -2 0 2 1 2],[ 3 0 1 2 3 3 3],[ 2 -1 0 1 2 2 2],[ 0 -2 -1 0 1 0 1],[-2 -3 -2 -1 0 -2 0],[-1 -3 -2 0 2 0 2],[-2 -3 -2 -1 0 -2 0]] |
| Primitive based matrix | [[ 0 2 2 1 0 -2 -3],[-2 0 0 -2 -1 -2 -3],[-2 0 0 -2 -1 -2 -3],[-1 2 2 0 0 -2 -3],[ 0 1 1 0 0 -1 -2],[ 2 2 2 2 1 0 -1],[ 3 3 3 3 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-1,0,2,3,0,2,1,2,3,2,1,2,3,0,2,3,1,2,1] |
| Phi over symmetry | [-3,-2,0,1,2,2,0,1,1,2,2,1,1,2,2,1,1,1,-1,-1,0] |
| Phi of -K | [-3,-2,0,1,2,2,0,1,1,2,2,1,1,2,2,1,1,1,-1,-1,0] |
| Phi of K* | [-2,-2,-1,0,2,3,0,-1,1,2,2,-1,1,2,2,1,1,1,1,1,0] |
| Phi of -K* | [-3,-2,0,1,2,2,1,2,3,3,3,1,2,2,2,0,1,1,2,2,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 3z+7 |
| Enhanced Jones-Krushkal polynomial | 12w^4z-14w^3z+5w^2z+7w |
| Inner characteristic polynomial | t^6+55t^4+11t^2 |
| Outer characteristic polynomial | t^7+77t^5+56t^3 |
| Flat arrow polynomial | -16*K1**4 + 4*K1**3 + 8*K1**2*K2 + 6*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -864*K1**4 - 1792*K1**2*K2**6 + 2176*K1**2*K2**5 - 3264*K1**2*K2**4 + 1184*K1**2*K2**3 - 2048*K1**2*K2**2 + 2512*K1**2*K2 - 192*K1**2*K3**2 - 1252*K1**2 + 1408*K1*K2**5*K3 + 1088*K1*K2**3*K3 + 1696*K1*K2*K3 + 224*K1*K3*K4 - 768*K2**8 + 512*K2**6*K4 - 1568*K2**6 - 320*K2**4*K3**2 - 64*K2**4*K4**2 + 544*K2**4*K4 + 520*K2**4 + 32*K2**3*K3*K5 - 240*K2**2*K3**2 - 40*K2**2*K4**2 + 376*K2**2*K4 - 520*K2**2 + 112*K2*K3*K5 - 440*K3**2 - 110*K4**2 - 20*K5**2 + 1116 |
| 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}], [{2, 6}, {5}, {4}, {3}, {1}], [{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}], [{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}, {2, 5}, {4}, {3}, {1}], [{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}, {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}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {1, 3}, {2}], [{6}, {5}, {4}, {2, 3}, {1}], [{6}, {5}, {4}, {3}, {1, 2}], [{6}, {5}, {4}, {3}, {2}, {1}]] |
| If K is slice | False |