| Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,0,1,1,2,4,1,0,1,2,0,0,1,-1,-1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.313'] |
| 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+86t^5+46t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.313'] |
| 2-strand cable arrow polynomial of the knot is: -336*K1**4 - 32*K1**3*K3 - 256*K1**2*K2**4 + 896*K1**2*K2**3 - 2944*K1**2*K2**2 - 96*K1**2*K2*K4 + 3280*K1**2*K2 - 16*K1**2*K3**2 - 2220*K1**2 + 576*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 640*K1*K2**2*K3 - 32*K1*K2**2*K5 + 2560*K1*K2*K3 + 104*K1*K3*K4 + 8*K1*K4*K5 - 64*K2**6 + 64*K2**4*K4 - 1096*K2**4 - 400*K2**2*K3**2 - 48*K2**2*K4**2 + 704*K2**2*K4 - 1008*K2**2 + 128*K2*K3*K5 + 8*K2*K4*K6 - 592*K3**2 - 86*K4**2 - 20*K5**2 + 1524 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.313'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11417', 'vk6.11704', 'vk6.12719', 'vk6.13072', 'vk6.20267', 'vk6.21584', 'vk6.27527', 'vk6.29103', 'vk6.31154', 'vk6.31487', 'vk6.32308', 'vk6.32745', 'vk6.38930', 'vk6.41150', 'vk6.45685', 'vk6.47404', 'vk6.52168', 'vk6.52399', 'vk6.52985', 'vk6.53312', 'vk6.57092', 'vk6.58255', 'vk6.61660', 'vk6.62828', 'vk6.63742', 'vk6.63842', 'vk6.64160', 'vk6.64356', 'vk6.66731', 'vk6.67600', 'vk6.69377', 'vk6.70113', 'vk6.81984', 'vk6.82716', 'vk6.84394', 'vk6.85986', 'vk6.85987', 'vk6.88173', 'vk6.88757', 'vk6.89120'] |
| 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 | O1O2O3O4O5U2U3O6U4U5U1U6 |
| R3 orbit | {'O1O2O3O4U1O5U3O6U4U2U5U6', 'O1O2O3O4O5U2U3O6U4U5U1U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U6U5U1U2O6U3U4 |
| Gauss code of K* | O1O2O3O4U3U5U6U1U2O5O6U4 |
| Gauss code of -K* | O1O2O3O4U1O5O6U3U4U5U6U2 |
| 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 -3 -1 0 2 3],[ 1 0 -3 -1 1 3 3],[ 3 3 0 1 2 3 2],[ 1 1 -1 0 1 2 2],[ 0 -1 -2 -1 0 1 2],[-2 -3 -3 -2 -1 0 1],[-3 -3 -2 -2 -2 -1 0]] |
| Primitive based matrix | [[ 0 3 2 0 -1 -1 -3],[-3 0 -1 -2 -2 -3 -2],[-2 1 0 -1 -2 -3 -3],[ 0 2 1 0 -1 -1 -2],[ 1 2 2 1 0 1 -1],[ 1 3 3 1 -1 0 -3],[ 3 2 3 2 1 3 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,0,1,1,3,1,2,2,3,2,1,2,3,3,1,1,2,-1,1,3] |
| Phi over symmetry | [-3,-2,0,1,1,3,0,1,1,2,4,1,0,1,2,0,0,1,-1,-1,1] |
| Phi of -K | [-3,-1,-1,0,2,3,-1,1,1,2,4,1,0,0,1,0,1,2,1,1,0] |
| Phi of K* | [-3,-2,0,1,1,3,0,1,1,2,4,1,0,1,2,0,0,1,-1,-1,1] |
| Phi of -K* | [-3,-1,-1,0,2,3,1,3,2,3,2,1,1,2,2,1,3,3,1,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 4z^2+17z+19 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+17w^2z+19w |
| Inner characteristic polynomial | t^6+62t^4+17t^2 |
| Outer characteristic polynomial | t^7+86t^5+46t^3+3t |
| Flat arrow polynomial | 8*K1**3 - 2*K1**2 - 4*K1*K2 - 4*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -336*K1**4 - 32*K1**3*K3 - 256*K1**2*K2**4 + 896*K1**2*K2**3 - 2944*K1**2*K2**2 - 96*K1**2*K2*K4 + 3280*K1**2*K2 - 16*K1**2*K3**2 - 2220*K1**2 + 576*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 640*K1*K2**2*K3 - 32*K1*K2**2*K5 + 2560*K1*K2*K3 + 104*K1*K3*K4 + 8*K1*K4*K5 - 64*K2**6 + 64*K2**4*K4 - 1096*K2**4 - 400*K2**2*K3**2 - 48*K2**2*K4**2 + 704*K2**2*K4 - 1008*K2**2 + 128*K2*K3*K5 + 8*K2*K4*K6 - 592*K3**2 - 86*K4**2 - 20*K5**2 + 1524 |
| 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}, {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}, {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 |