| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,-1,1,2,3,2,1,1,1,0,1,0,1,0,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.602'] |
| Arrow polynomial of the knot is: 8*K1**3 - 6*K1**2 - 6*K1*K2 - 3*K1 + 3*K2 + K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.237', '6.602', '6.956', '6.986', '6.992', '6.1052', '6.1059'] |
| Outer characteristic polynomial of the knot is: t^7+42t^5+50t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.602'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 - 256*K1**4*K2**2 + 2336*K1**4*K2 - 5104*K1**4 + 544*K1**3*K2*K3 + 64*K1**3*K3*K4 - 832*K1**3*K3 - 192*K1**2*K2**4 + 1120*K1**2*K2**3 + 224*K1**2*K2**2*K4 - 6880*K1**2*K2**2 - 896*K1**2*K2*K4 + 11024*K1**2*K2 - 592*K1**2*K3**2 - 144*K1**2*K4**2 - 6204*K1**2 + 256*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 1280*K1*K2**2*K3 - 64*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7976*K1*K2*K3 + 1576*K1*K3*K4 + 216*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 984*K2**4 - 256*K2**2*K3**2 - 72*K2**2*K4**2 + 1720*K2**2*K4 - 5546*K2**2 + 288*K2*K3*K5 + 32*K2*K4*K6 - 2532*K3**2 - 966*K4**2 - 144*K5**2 - 6*K6**2 + 5788 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.602'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4058', 'vk6.4089', 'vk6.5296', 'vk6.5327', 'vk6.7432', 'vk6.7457', 'vk6.8927', 'vk6.8958', 'vk6.10110', 'vk6.10277', 'vk6.10300', 'vk6.14561', 'vk6.15293', 'vk6.15420', 'vk6.15783', 'vk6.16200', 'vk6.29852', 'vk6.29883', 'vk6.33931', 'vk6.34009', 'vk6.34228', 'vk6.34391', 'vk6.48458', 'vk6.49158', 'vk6.50212', 'vk6.50239', 'vk6.51586', 'vk6.53974', 'vk6.54034', 'vk6.54183', 'vk6.54475', 'vk6.63299'] |
| 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 | O1O2O3O4U2O5U3O6U5U6U1U4 |
| R3 orbit | {'O1O2O3O4U2O5U3O6U5U6U1U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U4U5U6O5U2O6U3 |
| Gauss code of K* | O1O2O3O4U3U5U6U4O5U1O6U2 |
| Gauss code of -K* | O1O2O3O4U3O5U4O6U1U5U6U2 |
| 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 -1 -2 -1 3 0 1],[ 1 0 -1 0 3 0 1],[ 2 1 0 1 2 1 0],[ 1 0 -1 0 2 1 1],[-3 -3 -2 -2 0 -1 1],[ 0 0 -1 -1 1 0 1],[-1 -1 0 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 1 0 -1 -1 -2],[-3 0 1 -1 -2 -3 -2],[-1 -1 0 -1 -1 -1 0],[ 0 1 1 0 -1 0 -1],[ 1 2 1 1 0 0 -1],[ 1 3 1 0 0 0 -1],[ 2 2 0 1 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,1,1,2,-1,1,2,3,2,1,1,1,0,1,0,1,0,1,1] |
| Phi over symmetry | [-3,-1,0,1,1,2,-1,1,2,3,2,1,1,1,0,1,0,1,0,1,1] |
| Phi of -K | [-2,-1,-1,0,1,3,0,0,1,3,3,0,0,1,2,1,1,1,0,2,3] |
| Phi of K* | [-3,-1,0,1,1,2,3,2,1,2,3,0,1,1,3,1,0,1,0,0,0] |
| Phi of -K* | [-2,-1,-1,0,1,3,1,1,1,0,2,0,0,1,3,1,1,2,1,1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 3z^2+24z+37 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+24w^2z+37w |
| Inner characteristic polynomial | t^6+26t^4+25t^2+1 |
| Outer characteristic polynomial | t^7+42t^5+50t^3+8t |
| Flat arrow polynomial | 8*K1**3 - 6*K1**2 - 6*K1*K2 - 3*K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -64*K1**6 - 256*K1**4*K2**2 + 2336*K1**4*K2 - 5104*K1**4 + 544*K1**3*K2*K3 + 64*K1**3*K3*K4 - 832*K1**3*K3 - 192*K1**2*K2**4 + 1120*K1**2*K2**3 + 224*K1**2*K2**2*K4 - 6880*K1**2*K2**2 - 896*K1**2*K2*K4 + 11024*K1**2*K2 - 592*K1**2*K3**2 - 144*K1**2*K4**2 - 6204*K1**2 + 256*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 1280*K1*K2**2*K3 - 64*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7976*K1*K2*K3 + 1576*K1*K3*K4 + 216*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 984*K2**4 - 256*K2**2*K3**2 - 72*K2**2*K4**2 + 1720*K2**2*K4 - 5546*K2**2 + 288*K2*K3*K5 + 32*K2*K4*K6 - 2532*K3**2 - 966*K4**2 - 144*K5**2 - 6*K6**2 + 5788 |
| 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}, {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}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]] |
| If K is slice | False |