| Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,0,1,1,3,2,0,2,3,3,1,0,1,1,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.623'] |
| Arrow polynomial of the knot is: 8*K1**3 - 2*K1**2 - 4*K1*K2 - 4*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.313', '6.623', '6.1031', '6.1201', '6.1327', '6.1378', '6.1640', '6.1697', '6.1797', '6.1833'] |
| Outer characteristic polynomial of the knot is: t^7+72t^5+312t^3+12t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.623'] |
| 2-strand cable arrow polynomial of the knot is: -208*K1**4 - 32*K1**3*K3 + 384*K1**2*K2**5 - 1920*K1**2*K2**4 + 4096*K1**2*K2**3 + 32*K1**2*K2**2*K4 - 7136*K1**2*K2**2 - 224*K1**2*K2*K4 + 6176*K1**2*K2 - 16*K1**2*K3**2 - 4148*K1**2 + 1440*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 2208*K1*K2**2*K3 - 160*K1*K2**2*K5 - 96*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 4920*K1*K2*K3 + 272*K1*K3*K4 + 8*K1*K4*K5 - 576*K2**6 + 224*K2**4*K4 - 3080*K2**4 - 464*K2**2*K3**2 - 176*K2**2*K4**2 + 2376*K2**2*K4 - 1312*K2**2 + 152*K2*K3*K5 + 48*K2*K4*K6 - 1024*K3**2 - 286*K4**2 - 4*K5**2 + 2780 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.623'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71621', 'vk6.71783', 'vk6.72204', 'vk6.72350', 'vk6.73374', 'vk6.73537', 'vk6.75283', 'vk6.75550', 'vk6.77243', 'vk6.77327', 'vk6.77576', 'vk6.77684', 'vk6.78270', 'vk6.78520', 'vk6.80082', 'vk6.80232', 'vk6.81120', 'vk6.81167', 'vk6.81190', 'vk6.81245', 'vk6.81327', 'vk6.81514', 'vk6.82015', 'vk6.82416', 'vk6.82751', 'vk6.85451', 'vk6.86347', 'vk6.86926', 'vk6.87146', 'vk6.88084', 'vk6.88669', 'vk6.88765'] |
| 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 | O1O2O3O4U5O6U2O5U1U6U3U4 |
| R3 orbit | {'O1O2O3O4U5O6U2O5U1U6U3U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U2U5U4O6U3O5U6 |
| Gauss code of K* | O1O2O3O4U1U5U3U4O6U2O5U6 |
| Gauss code of -K* | O1O2O3O4U5O6U3O5U1U2U6U4 |
| 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 1 3 0 1],[ 3 0 1 3 4 2 1],[ 2 -1 0 1 2 2 0],[-1 -3 -1 0 1 0 -1],[-3 -4 -2 -1 0 -2 -1],[ 0 -2 -2 0 2 0 1],[-1 -1 0 1 1 -1 0]] |
| Primitive based matrix | [[ 0 3 1 1 0 -2 -3],[-3 0 -1 -1 -2 -2 -4],[-1 1 0 1 -1 0 -1],[-1 1 -1 0 0 -1 -3],[ 0 2 1 0 0 -2 -2],[ 2 2 0 1 2 0 -1],[ 3 4 1 3 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,-1,0,2,3,1,1,2,2,4,-1,1,0,1,0,1,3,2,2,1] |
| Phi over symmetry | [-3,-2,0,1,1,3,0,1,1,3,2,0,2,3,3,1,0,1,1,1,1] |
| Phi of -K | [-3,-2,0,1,1,3,0,1,1,3,2,0,2,3,3,1,0,1,1,1,1] |
| Phi of K* | [-3,-1,-1,0,2,3,1,1,1,3,2,-1,1,2,1,0,3,3,0,1,0] |
| Phi of -K* | [-3,-2,0,1,1,3,1,2,1,3,4,2,0,1,2,1,0,2,1,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 4z^2+17z+19 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+8w^3z^2-4w^3z+21w^2z+19w |
| Inner characteristic polynomial | t^6+48t^4+159t^2+1 |
| Outer characteristic polynomial | t^7+72t^5+312t^3+12t |
| Flat arrow polynomial | 8*K1**3 - 2*K1**2 - 4*K1*K2 - 4*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -208*K1**4 - 32*K1**3*K3 + 384*K1**2*K2**5 - 1920*K1**2*K2**4 + 4096*K1**2*K2**3 + 32*K1**2*K2**2*K4 - 7136*K1**2*K2**2 - 224*K1**2*K2*K4 + 6176*K1**2*K2 - 16*K1**2*K3**2 - 4148*K1**2 + 1440*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 2208*K1*K2**2*K3 - 160*K1*K2**2*K5 - 96*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 4920*K1*K2*K3 + 272*K1*K3*K4 + 8*K1*K4*K5 - 576*K2**6 + 224*K2**4*K4 - 3080*K2**4 - 464*K2**2*K3**2 - 176*K2**2*K4**2 + 2376*K2**2*K4 - 1312*K2**2 + 152*K2*K3*K5 + 48*K2*K4*K6 - 1024*K3**2 - 286*K4**2 - 4*K5**2 + 2780 |
| 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}], [{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}], [{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}, {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}, {4}, {1, 3}, {2}]] |
| If K is slice | False |