| Min(phi) over symmetries of the knot is: [-4,-2,1,1,2,2,0,1,2,4,5,0,0,1,2,0,0,0,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.397'] |
| Arrow polynomial of the knot is: 4*K1**2*K2 - 2*K1*K3 - K2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.64', '6.74', '6.106', '6.178', '6.300', '6.397', '6.479', '6.481', '6.500'] |
| Outer characteristic polynomial of the knot is: t^7+81t^5+107t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.397'] |
| 2-strand cable arrow polynomial of the knot is: -512*K1**4 + 384*K1**3*K2*K3 - 1664*K1**2*K2**2 - 608*K1**2*K2*K4 + 2200*K1**2*K2 - 832*K1**2*K3**2 - 2768*K1**2 + 256*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 512*K1*K2**2*K3 - 32*K1*K2**2*K5 - 96*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 4496*K1*K2*K3 - 32*K1*K2*K4*K5 + 2176*K1*K3*K4 + 48*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 544*K2**4 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 736*K2**2*K3**2 - 304*K2**2*K4**2 + 936*K2**2*K4 - 8*K2**2*K6**2 - 2184*K2**2 - 320*K2*K3**2*K4 + 520*K2*K3*K5 + 368*K2*K4*K6 + 144*K3**2*K6 - 2208*K3**2 - 1034*K4**2 - 128*K5**2 - 136*K6**2 + 2808 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.397'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17138', 'vk6.17381', 'vk6.20598', 'vk6.22008', 'vk6.23547', 'vk6.23885', 'vk6.28064', 'vk6.29519', 'vk6.35715', 'vk6.36134', 'vk6.39478', 'vk6.41681', 'vk6.43046', 'vk6.43352', 'vk6.46067', 'vk6.47730', 'vk6.55283', 'vk6.55531', 'vk6.57474', 'vk6.58636', 'vk6.59709', 'vk6.60051', 'vk6.62147', 'vk6.63107', 'vk6.65092', 'vk6.65278', 'vk6.67002', 'vk6.67867', 'vk6.68338', 'vk6.68486', 'vk6.69620', 'vk6.70310'] |
| 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 | O1O2O3O4O5U4U3O6U2U1U6U5 |
| R3 orbit | {'O1O2O3O4O5U4U3O6U2U1U6U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U1U6U5U4O6U3U2 |
| Gauss code of K* | O1O2O3O4U2U1U5U6U4O6O5U3 |
| Gauss code of -K* | O1O2O3O4U2O5O6U1U6U5U4U3 |
| 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 -2 -2 -1 -1 4 2],[ 2 0 0 0 0 5 2],[ 2 0 0 0 0 4 1],[ 1 0 0 0 0 2 0],[ 1 0 0 0 0 1 0],[-4 -5 -4 -2 -1 0 0],[-2 -2 -1 0 0 0 0]] |
| Primitive based matrix | [[ 0 4 2 -1 -1 -2 -2],[-4 0 0 -1 -2 -4 -5],[-2 0 0 0 0 -1 -2],[ 1 1 0 0 0 0 0],[ 1 2 0 0 0 0 0],[ 2 4 1 0 0 0 0],[ 2 5 2 0 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-4,-2,1,1,2,2,0,1,2,4,5,0,0,1,2,0,0,0,0,0,0] |
| Phi over symmetry | [-4,-2,1,1,2,2,0,1,2,4,5,0,0,1,2,0,0,0,0,0,0] |
| Phi of -K | [-2,-2,-1,-1,2,4,0,1,1,2,1,1,1,3,2,0,3,3,3,4,2] |
| Phi of K* | [-4,-2,1,1,2,2,2,3,4,1,2,3,3,2,3,0,1,1,1,1,0] |
| Phi of -K* | [-2,-2,-1,-1,2,4,0,0,0,1,4,0,0,2,5,0,0,1,0,2,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^4+t^2+2t |
| Normalized Jones-Krushkal polynomial | 9z^2+30z+25 |
| Enhanced Jones-Krushkal polynomial | 9w^3z^2+30w^2z+25w |
| Inner characteristic polynomial | t^6+51t^4+34t^2 |
| Outer characteristic polynomial | t^7+81t^5+107t^3+7t |
| Flat arrow polynomial | 4*K1**2*K2 - 2*K1*K3 - K2 |
| 2-strand cable arrow polynomial | -512*K1**4 + 384*K1**3*K2*K3 - 1664*K1**2*K2**2 - 608*K1**2*K2*K4 + 2200*K1**2*K2 - 832*K1**2*K3**2 - 2768*K1**2 + 256*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 512*K1*K2**2*K3 - 32*K1*K2**2*K5 - 96*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 4496*K1*K2*K3 - 32*K1*K2*K4*K5 + 2176*K1*K3*K4 + 48*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 544*K2**4 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 736*K2**2*K3**2 - 304*K2**2*K4**2 + 936*K2**2*K4 - 8*K2**2*K6**2 - 2184*K2**2 - 320*K2*K3**2*K4 + 520*K2*K3*K5 + 368*K2*K4*K6 + 144*K3**2*K6 - 2208*K3**2 - 1034*K4**2 - 128*K5**2 - 136*K6**2 + 2808 |
| 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}], [{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}], [{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}, {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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {1, 3}, {2}]] |
| If K is slice | False |