| Min(phi) over symmetries of the knot is: [-3,-3,0,1,2,3,-1,1,2,4,3,1,2,3,2,0,1,1,1,2,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.212'] |
| Arrow polynomial of the knot is: 12*K1**3 - 6*K1**2 - 6*K1*K2 - 6*K1 + 3*K2 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.212', '6.358', '6.588', '6.589', '6.603', '6.990'] |
| Outer characteristic polynomial of the knot is: t^7+92t^5+45t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.212'] |
| 2-strand cable arrow polynomial of the knot is: 96*K1**4*K2 - 400*K1**4 + 320*K1**2*K2**3 - 2176*K1**2*K2**2 - 192*K1**2*K2*K4 + 3488*K1**2*K2 - 16*K1**2*K3**2 - 2856*K1**2 + 160*K1*K2**3*K3 - 736*K1*K2**2*K3 - 64*K1*K2**2*K5 + 3024*K1*K2*K3 + 400*K1*K3*K4 + 24*K1*K4*K5 - 96*K2**6 + 96*K2**4*K4 - 888*K2**4 - 224*K2**2*K3**2 - 104*K2**2*K4**2 + 1288*K2**2*K4 - 2180*K2**2 + 160*K2*K3*K5 + 80*K2*K4*K6 - 968*K3**2 - 442*K4**2 - 40*K5**2 - 20*K6**2 + 2264 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.212'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71598', 'vk6.71614', 'vk6.71723', 'vk6.71736', 'vk6.72139', 'vk6.72155', 'vk6.72335', 'vk6.73767', 'vk6.73802', 'vk6.73905', 'vk6.73936', 'vk6.75745', 'vk6.75906', 'vk6.75915', 'vk6.77213', 'vk6.77230', 'vk6.77521', 'vk6.77538', 'vk6.78700', 'vk6.78712', 'vk6.78754', 'vk6.78898', 'vk6.78913', 'vk6.79051', 'vk6.79617', 'vk6.80321', 'vk6.80330', 'vk6.80355', 'vk6.80444', 'vk6.80453', 'vk6.80571', 'vk6.81022', 'vk6.81365', 'vk6.81726', 'vk6.84467', 'vk6.85419', 'vk6.87985', 'vk6.88347', 'vk6.88349', 'vk6.89318'] |
| 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 | O1O2O3O4O5U2O6U5U1U4U6U3 |
| R3 orbit | {'O1O2O3O4U1O5O6U2U5U4U6U3', 'O1O2O3O4O5U2O6U5U1U4U6U3'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U3U6U2U5U1O6U4 |
| Gauss code of K* | O1O2O3O4O5U2U6U5U3U1O6U4 |
| Gauss code of -K* | O1O2O3O4O5U2O6U5U3U1U6U4 |
| 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 -3 2 1 0 3],[ 3 0 -1 4 2 1 3],[ 3 1 0 3 2 1 2],[-2 -4 -3 0 -1 -1 2],[-1 -2 -2 1 0 0 2],[ 0 -1 -1 1 0 0 1],[-3 -3 -2 -2 -2 -1 0]] |
| Primitive based matrix | [[ 0 3 2 1 0 -3 -3],[-3 0 -2 -2 -1 -2 -3],[-2 2 0 -1 -1 -3 -4],[-1 2 1 0 0 -2 -2],[ 0 1 1 0 0 -1 -1],[ 3 2 3 2 1 0 1],[ 3 3 4 2 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-1,0,3,3,2,2,1,2,3,1,1,3,4,0,2,2,1,1,-1] |
| Phi over symmetry | [-3,-3,0,1,2,3,-1,1,2,4,3,1,2,3,2,0,1,1,1,2,2] |
| Phi of -K | [-3,-3,0,1,2,3,-1,2,2,2,4,2,2,1,3,1,1,2,0,0,-1] |
| Phi of K* | [-3,-2,-1,0,3,3,-1,0,2,3,4,0,1,1,2,1,2,2,2,2,-1] |
| Phi of -K* | [-3,-3,0,1,2,3,-1,1,2,4,3,1,2,3,2,0,1,1,1,2,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+16w^2z+21w |
| Inner characteristic polynomial | t^6+60t^4+8t^2 |
| Outer characteristic polynomial | t^7+92t^5+45t^3+3t |
| Flat arrow polynomial | 12*K1**3 - 6*K1**2 - 6*K1*K2 - 6*K1 + 3*K2 + 4 |
| 2-strand cable arrow polynomial | 96*K1**4*K2 - 400*K1**4 + 320*K1**2*K2**3 - 2176*K1**2*K2**2 - 192*K1**2*K2*K4 + 3488*K1**2*K2 - 16*K1**2*K3**2 - 2856*K1**2 + 160*K1*K2**3*K3 - 736*K1*K2**2*K3 - 64*K1*K2**2*K5 + 3024*K1*K2*K3 + 400*K1*K3*K4 + 24*K1*K4*K5 - 96*K2**6 + 96*K2**4*K4 - 888*K2**4 - 224*K2**2*K3**2 - 104*K2**2*K4**2 + 1288*K2**2*K4 - 2180*K2**2 + 160*K2*K3*K5 + 80*K2*K4*K6 - 968*K3**2 - 442*K4**2 - 40*K5**2 - 20*K6**2 + 2264 |
| 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 |