| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,1,1,3,2,1,1,1,1,-1,0,0,-1,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.994'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['4.2', '6.303', '6.338', '6.381', '6.432', '6.468', '6.558', '6.583', '6.597', '6.607', '6.634', '6.637', '6.643', '6.654', '6.667', '6.701', '6.709', '6.712', '6.718', '6.728', '6.767', '6.801', '6.825', '6.827', '6.974', '6.994', '6.1042', '6.1061', '6.1069', '6.1181', '6.1271', '6.1286', '6.1287', '6.1289', '6.1297', '6.1337', '6.1355'] |
| Outer characteristic polynomial of the knot is: t^7+60t^5+72t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.994'] |
| 2-strand cable arrow polynomial of the knot is: -208*K1**4 - 32*K1**3*K3 + 512*K1**2*K2**5 - 1280*K1**2*K2**4 + 3424*K1**2*K2**3 + 32*K1**2*K2**2*K4 - 5872*K1**2*K2**2 - 224*K1**2*K2*K4 + 4960*K1**2*K2 - 16*K1**2*K3**2 - 3452*K1**2 - 1024*K1*K2**4*K3 + 1568*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1056*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 3472*K1*K2*K3 + 272*K1*K3*K4 - 864*K2**6 + 864*K2**4*K4 - 2104*K2**4 - 480*K2**2*K3**2 - 88*K2**2*K4**2 + 968*K2**2*K4 - 776*K2**2 + 64*K2*K3*K5 - 724*K3**2 - 162*K4**2 + 2160 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.994'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71623', 'vk6.71786', 'vk6.72208', 'vk6.72352', 'vk6.73375', 'vk6.73536', 'vk6.75284', 'vk6.75549', 'vk6.77241', 'vk6.77323', 'vk6.77573', 'vk6.77682', 'vk6.78271', 'vk6.78519', 'vk6.80083', 'vk6.80231', 'vk6.81115', 'vk6.81171', 'vk6.81192', 'vk6.81239', 'vk6.81338', 'vk6.81526', 'vk6.82009', 'vk6.82419', 'vk6.82740', 'vk6.85449', 'vk6.86345', 'vk6.86928', 'vk6.87150', 'vk6.88092', 'vk6.88655', 'vk6.88761'] |
| 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 | O1O2O3O4U1U5O6U4O5U2U3U6 |
| R3 orbit | {'O1O2O3O4U1U5O6U4O5U2U3U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U2U3O6U1O5U6U4 |
| Gauss code of K* | O1O2O3U4U1U2U5O4O6U3O5U6 |
| Gauss code of -K* | O1O2O3U4O5U1O4O6U5U2U3U6 |
| 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 -1 1 1 0 2],[ 3 0 2 3 1 2 3],[ 1 -2 0 1 1 0 2],[-1 -3 -1 0 1 -2 1],[-1 -1 -1 -1 0 -1 0],[ 0 -2 0 2 1 0 2],[-2 -3 -2 -1 0 -2 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -1 -3],[-2 0 0 -1 -2 -2 -3],[-1 0 0 -1 -1 -1 -1],[-1 1 1 0 -2 -1 -3],[ 0 2 1 2 0 0 -2],[ 1 2 1 1 0 0 -2],[ 3 3 1 3 2 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,1,3,0,1,2,2,3,1,1,1,1,2,1,3,0,2,2] |
| Phi over symmetry | [-3,-1,0,1,1,2,0,1,1,3,2,1,1,1,1,-1,0,0,-1,0,1] |
| Phi of -K | [-3,-1,0,1,1,2,0,1,1,3,2,1,1,1,1,-1,0,0,-1,0,1] |
| Phi of K* | [-2,-1,-1,0,1,3,0,1,0,1,2,1,-1,1,1,0,1,3,1,1,0] |
| Phi of -K* | [-3,-1,0,1,1,2,2,2,1,3,3,0,1,1,2,1,2,2,-1,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| 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+44t^4+23t^2+1 |
| Outer characteristic polynomial | t^7+60t^5+72t^3+8t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -208*K1**4 - 32*K1**3*K3 + 512*K1**2*K2**5 - 1280*K1**2*K2**4 + 3424*K1**2*K2**3 + 32*K1**2*K2**2*K4 - 5872*K1**2*K2**2 - 224*K1**2*K2*K4 + 4960*K1**2*K2 - 16*K1**2*K3**2 - 3452*K1**2 - 1024*K1*K2**4*K3 + 1568*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1056*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 3472*K1*K2*K3 + 272*K1*K3*K4 - 864*K2**6 + 864*K2**4*K4 - 2104*K2**4 - 480*K2**2*K3**2 - 88*K2**2*K4**2 + 968*K2**2*K4 - 776*K2**2 + 64*K2*K3*K5 - 724*K3**2 - 162*K4**2 + 2160 |
| 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}], [{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}, {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 |