| Min(phi) over symmetries of the knot is: [-2,0,1,1,1,0,2,1,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1746', '6.1760', '7.43063'] |
| Arrow polynomial of the knot is: -10*K1**2 - 8*K1*K2 + 4*K1 + 5*K2 + 4*K3 + 6 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.372', '6.930', '6.1007', '6.1701', '6.1714', '6.1760', '6.1788'] |
| Outer characteristic polynomial of the knot is: t^5+19t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1746', '6.1760', '7.37934'] |
| 2-strand cable arrow polynomial of the knot is: -640*K1**6 - 320*K1**4*K2**2 + 1184*K1**4*K2 - 2832*K1**4 + 352*K1**3*K2*K3 - 2272*K1**2*K2**2 + 3992*K1**2*K2 - 1136*K1**2*K3**2 - 416*K1**2*K4**2 - 1384*K1**2 + 3000*K1*K2*K3 + 1336*K1*K3*K4 + 400*K1*K4*K5 - 296*K2**4 - 320*K2**2*K3**2 - 128*K2**2*K4**2 + 392*K2**2*K4 - 1808*K2**2 + 336*K2*K3*K5 + 128*K2*K4*K6 - 1096*K3**2 - 522*K4**2 - 184*K5**2 - 32*K6**2 + 2264 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1760'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4448', 'vk6.4543', 'vk6.5834', 'vk6.5961', 'vk6.7892', 'vk6.8006', 'vk6.9321', 'vk6.9440', 'vk6.13406', 'vk6.13503', 'vk6.13694', 'vk6.14064', 'vk6.15037', 'vk6.15157', 'vk6.17789', 'vk6.17820', 'vk6.18841', 'vk6.19434', 'vk6.19729', 'vk6.24336', 'vk6.25436', 'vk6.25467', 'vk6.26608', 'vk6.33260', 'vk6.33321', 'vk6.37560', 'vk6.44883', 'vk6.48635', 'vk6.50539', 'vk6.53644', 'vk6.55820', 'vk6.65492'] |
| 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 | O1O2O3U4U5O4O6U2U6O5U1U3 |
| R3 orbit | {'O1O2O3U4U5O4O6U2U6O5U1U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U3O4U5U2O5O6U4U6 |
| Gauss code of K* | O1O2U3O4O5U4U1U5O6O3U6U2 |
| Gauss code of -K* | O1O2U3O4O5U4U6O3O6U1U5U2 |
| 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 -1 -1 2 -1 0 1],[ 1 0 0 2 0 1 1],[ 1 0 0 1 1 1 1],[-2 -2 -1 0 -3 -1 0],[ 1 0 -1 3 0 0 1],[ 0 -1 -1 1 0 0 1],[-1 -1 -1 0 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 0 -1 -1],[-2 0 -1 -1 -3],[ 0 1 0 -1 0],[ 1 1 1 0 1],[ 1 3 0 -1 0]] |
| If based matrix primitive | False |
| Phi of primitive based matrix | [-2,0,1,1,1,1,3,1,0,-1] |
| Phi over symmetry | [-2,0,1,1,1,0,2,1,0,-1] |
| Phi of -K | [-1,-1,0,2,-1,0,2,1,0,1] |
| Phi of K* | [-2,0,1,1,1,0,2,1,0,-1] |
| Phi of -K* | [-1,-1,0,2,-1,0,3,1,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 15z+31 |
| Enhanced Jones-Krushkal polynomial | 15w^2z+31w |
| Inner characteristic polynomial | t^4+13t^2+4 |
| Outer characteristic polynomial | t^5+19t^3+7t |
| Flat arrow polynomial | -10*K1**2 - 8*K1*K2 + 4*K1 + 5*K2 + 4*K3 + 6 |
| 2-strand cable arrow polynomial | -640*K1**6 - 320*K1**4*K2**2 + 1184*K1**4*K2 - 2832*K1**4 + 352*K1**3*K2*K3 - 2272*K1**2*K2**2 + 3992*K1**2*K2 - 1136*K1**2*K3**2 - 416*K1**2*K4**2 - 1384*K1**2 + 3000*K1*K2*K3 + 1336*K1*K3*K4 + 400*K1*K4*K5 - 296*K2**4 - 320*K2**2*K3**2 - 128*K2**2*K4**2 + 392*K2**2*K4 - 1808*K2**2 + 336*K2*K3*K5 + 128*K2*K4*K6 - 1096*K3**2 - 522*K4**2 - 184*K5**2 - 32*K6**2 + 2264 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{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}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}]] |
| If K is slice | False |